Exponential loss of memory for the 2-dimensional Allen–Cahn equation with small noise

Abstract

We prove an asymptotic coupling theorem for the 2-dimensional Allen–Cahn equation perturbed by a small space-time white noise. We show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology. In the 1-dimensional case a similar result was shown in Martinelli et al. (Commun Math Phys 120(1):25–69, 1988; J Stat Phys 55(3–4):477–504, 1989). It is well-known that in two or more dimensions solutions of this equation are distribution-valued, and the equation has to be interpreted in a renormalised sense. Formally, this renormalisation corresponds to moving the minima of the potential infinitely far apart and making them infinitely deep. We show that despite this renormalisation, solutions behave like perturbations of the deterministic system without renormalisation: they spend large stretches of time close to the minimisers of the (un-renormalised) potential and the exponential contraction rate of different profiles is given by the second derivative of the potential in these points. As an application we prove an Eyring–Kramers law for the transition times between the stable solutions of the deterministic system for fixed initial conditions.

1 Introduction

We are interested in the behaviour of solutions to the Allen–Cahn equation, perturbed by a small noise term. The deterministic equation is given by

$$\begin{aligned} (\partial _t-\Delta ) X = - X^3 + X, \end{aligned}$$

(1.1)

and it is well-known that (1.1) is a gradient flow with respect to the potential

$$\begin{aligned} V(X) := \int \left( \frac{1}{2}|\nabla X(z)|^2 - \frac{1}{2} X(z)^2 + \frac{1}{4} X(z)^4 \right) \,\mathrm {d}z. \end{aligned}$$

(1.2)

The fluctuation-dissipation theorem suggests an additive Gaussian space-time white noise $\xi$ as a natural random perturbation of (1.1); so we consider

$$\begin{aligned} (\partial _t - \Delta ) X = -X^3 + X + \sqrt{2\varepsilon } \xi , \end{aligned}$$

(1.3)

for a small parameter $\varepsilon >0$.

In the 1-dimensional case, i.e. the case where the solution X depends on time and a 1-dimensional spatial argument, the behaviour of solutions to (1.3) is well-understood. Solutions exhibit the phenomenon of metastability, i.e. they typically spend large stretches of time close to the minimisers of the potential (1.2) with rare and relatively quick noise-induced transitions between them. Early contributions go back to the 80s where Farris and Jona–Lasinio [9] studied the system on the level of large deviations.

We are particularly interested in the “exponential loss of memory property” first observed by Martinelli, Olivieri and Scoppola in [17, 18]. They studied the flow map induced by (1.3), i.e. the random map $x \mapsto X(t;x)$ which associates to any initial condition the corresponding solution at time t, and showed that for large t the map becomes essentially constant. They also showed that with overwhelming probability, solutions that start within the basin of attraction of the same minimiser of V contract exponentially fast, with exponential rate given by the smallest eigenvalue of the linearisation of V in this minimiser. This implies for example that the law of such solutions at large times is essentially insensitive to the precise location at which they are started.

It is very natural to consider higher dimensional analogues of (1.3), but unfortunately for space dimension $d \ge 2$, Eq. (1.3) is ill-posed. In fact, for $d \ge 2$ the space-time white noise becomes so irregular, that solutions have to be interpreted in the sense of Schwartz distributions, and the interpretation of the nonlinear term is a priori unclear. These kind of singular stochastic partial differential equations (SPDEs) have received a lot of attention recently (see e.g. [6, 10, 12]). The solution proposed in these works is to renormalise the equation, by removing some infinite terms, formally leading to the equation

$$\begin{aligned} (\partial _t - \Delta ) X = - X^3 + (1+3\varepsilon \infty ) X + \sqrt{2\varepsilon } \xi . \end{aligned}$$

(1.4)

Note that formally, this renormalisation corresponds to moving the minima of the double-well potential out to $\pm \infty$ and making them infinitely deep at the same time. So at first glance, it seems unclear why these renormalised distribution-valued solutions should exhibit similar behaviour to the 1-dimensional function-valued solutions of (1.3).

In [14] Hairer and the second named author studied the small $\varepsilon$ asymptotics for (1.4) for space dimension $d=2$ and 3 on the level of Freidlin–Wentzell type large deviations. They obtained a large deviation principle with rate function $\mathcal {I}$ given by

$$\begin{aligned} \mathcal {I}(X) := \frac{1}{4} \int _0^T \int \left( \partial _t X(t,z) - \left( \Delta X(t,z) -\left( X(t,z)^3 - X(t,z) \right) \right) \right) ^2 \,\mathrm {d}z \,\mathrm {d}t. \end{aligned}$$

(1.5)

In fact, a result in a similar spirit had already appeared in the 90s [15]. The striking fact is that this rate function is exactly the 2-dimensional version of the rate function obtained in the 1-dimensional case [9]. The infinite renormalisation constant does not affect the rate functional. This result implies that for small $\varepsilon$ solutions of the renormalised SPDE (1.4) stay close to solutions of the deterministic PDE (1.1) suggesting that (1.4) may indeed be the natural small noise perturbation of (1.1).

In this article we consider (1.4) over a 2-dimensional torus $\mathbb {T}^2 = \mathbb {R}^2/L\mathbb {Z}^2$ for $L<2 \pi$. It is known that under this assumption on the torus size L, the deterministic equation (1.1) has exactly three stationary solutions, namely the constant profiles $-1,0,1$ (see [16, Appendix B.1]). Here $\pm 1$ are stable minimisers of V and 0 is unstable. We prove that in the small noise regime with overwhelming probability solutions that start close to the same stable minimiser $\pm 1$ contract exponentially fast. The exponential contraction rate is arbitrarily close to 2, the second derivative of the double-well $x \mapsto \frac{1}{4} x^4 - \frac{1}{2} x^2$ in $\pm 1$. This is precisely the 2-dimensional version of [17, Corollary 3.1].

On a technical level we work with the Da Prato–Debussche decomposition (see Sect. 2 for more details). An immediate observation is that differences of any two profiles have much better regularity than the solutions themselves. We split the time axis into random “good” and “bad” intervals depending on whether a reference profile is close to $\pm 1$ or not. The key idea is that on “good” intervals solutions should contract exponentially, while they should not diverge too fast on “bad” intervals. Furthermore, “good” intervals should be much longer than “bad” intervals.

The control on the “good” intervals is relatively straightforward: the exponential contraction follows by linearising the equation and the fact that these intervals are typically long follows from exponential moment bounds on the explicit stochastic objects appearing in the Da Prato–Debussche approach. The control on the “bad” intervals is much more involved: in the 1-dimensional case two profiles cannot diverge too fast, because the second derivative of the double-well potential is bounded from below. But in the 2-dimensional case, where solutions are distribution-valued, there is no obvious counterpart of this property. Instead we use a strong a priori estimate obtained in our previous work [24] and the local Lipschitz continuity of the non-linearity. Ultimately this yields an exponential growth bound where the exponential rate is given by a polynomial in the explicit stochastic objects. We use a large deviation estimate to prove that these intervals cannot be too long. In the final step we show that the exponential contraction holds for all t if a certain random walk with positive drift stays positive for all times. This random walk is then analysed using techniques developed for the classical Cramér–Lunberg model in risk theory.

The original motivation for our work was to prove an Eyring–Kramers law for the transition times of X. In [2] Berglund, Di Gesú and the second named author studied spectral Galerkin approximations $X_N$ of (1.4) and obtained explicit estimates on the expected first transition times from a neighbourhood of $-1$ to a neighbourhood of 1. These estimates give a precise asymptotic as $\varepsilon \rightarrow 0$ and hold uniformly in the discretisation parameter N. Their method was based on the potential theoretic approach developed in the finite-dimensional context by Bovier et al. in [3]. This approach relies heavily on the reversibility of the dynamics and provides explicit formulas for the expected transition times in terms of certain integrals of the reversible measure. The key observation in [2] was that in the context of (1.3) these integrals can be analysed uniformly in the parameter N using the classical Nelson’s estimate [22] from constructive Quantum Field Theory. However, the result in [2] was not optimal for the following two reasons: First, it does not allow to pass to the limit as $N\rightarrow \infty$ to retrieve the estimate for the transition times of X. Second, and more important, the bounds could only be obtained for a certain N-dependent choice of initial distribution on the neighbourhood of $-1$. This problem is inherent to the potential theoretic approach, which only yields an exact formula for the diffusion started in this so-called normalised equilibrium measure. In fact, a large part of the original work [3] was dedicated to removing this problem using regularity theory for the finite-dimensional transition probabilities.

In this paper we overcome these two barriers. We first justify the passage to the limit $N\rightarrow \infty$ based on our previous work [24]: we use the strong a priori estimates on the level of the approximation $X_N$ and the support theorem obtained there to prove uniform integrability of the transition times of $X_N$. The only difficulty here comes from the action of the Galerkin projection on the non-linearity which does not allow to test the equation with powers greater than 1. To remove the unnatural assumption on the initial distribution we make use of our main result, the exponential contraction estimate. This estimate allows us to couple the solution started with an arbitrary but fixed initial condition with the solution started in the normalised equilibrium measure.

1.1 Outline

In Sect. 2 we briefly review the solution theory of (1.4). In Sect. 3 we state our main results, that is, the exponential loss of memory, Theorem 3.1, and the Eyring–Kramers law, Theorem 3.5. In Sect. 4 we prove Theorem 3.1 based on some auxiliary propositions. These propositions are proved in Sects. 5 and 6. Finally, in Sect. 7 we prove the Eyring–Kramers law, Theorem 3.5, generalising [2, Theorem 2.3]. Several known results that are used throughout this article as well as some additional technical statements can be found in the Appendix.

1.2 Notation

We fix a torus $\mathbb {T}^2 = \mathbb {R}^2/L\mathbb {Z}^2$ of size $0<L<2\pi$. All function spaces are defined over $\mathbb {T}^2$. We write $\mathcal {C}^\infty$ for the space of smooth functions and $L^p, p\in [1,\infty ]$, for the space of p-integrable periodic functions endowed with the usual norm $\Vert \cdot \Vert _{L^p}$ and the usual interpretation if $p=\infty$.

We denote by $\mathcal {B}^\alpha _{p,q}$ the (inhomogeneous) Besov space of regularity $\alpha$ and exponents $p,q\in [1,\infty ]$ with norm $\Vert \cdot \Vert _{\mathcal {B}^\alpha _{p,q}}$ (see Definition A.1). We write $\mathcal {C}^\alpha$ and $\Vert \cdot \Vert _{\mathcal {C}^\alpha }$ to denote the space $\mathcal {B}^\alpha _{\infty ,\infty }$ and the corresponding norm. Many useful results about Besov spaces that we repeatedly use throughout the article can be found in Appendix A.

For any Banach space $(V, \Vert \cdot \Vert _V)$ we denote by $B_V(x_0;\delta )$ the open ball $\{x\in V: \Vert x-x_0\Vert _V < \delta \}$ and by ${\bar{B}}_V(x_0;\delta )$ its closure.

Throughout this article we write C for a positive constant which might change from line to line. In proofs we sometimes write $\lesssim$ instead of $\le C$. We also write $a\vee b$ and $a\wedge b$ to denote the maximum and the minimum of a and b.

In several statements we write $\pm 1$ to signify that the statement holds true for either choice of $+1$ or $-1$.

2 Preliminaries

Fix a probability space $(\Omega ,\mathcal {F},\mathbb {P})$ and let $\xi$ be a space-time white noise defined over $\Omega$. More precisely, $\xi$ is a family $\{\xi (\phi )\}_{\phi \in L^2((0,\infty )\times \mathbb {T}^2)}$ of centred Gaussian random variables such that

$$\begin{aligned} \mathbb {E}\xi (\phi )\xi (\psi ) = \langle \phi ,\psi \rangle _{L^2((0,\infty )\times \mathbb {T}^2)}. \end{aligned}$$

A natural filtration $\{\mathcal {F}_t\}_{t\ge 0}$ is given by the usual augmentation (as in [23, Chapter 1.4]) of

$$\begin{aligned} \tilde{\mathcal {F}}_t = \sigma \left( \{\xi (\phi ):\phi |_{(t,\infty )\times \mathbb {T}^2}\equiv 0 \} \right) , \quad t\ge 0. \end{aligned}$$

We interpret solutions of (1.4) following [6] and [20]. We write $X( \cdot ;x)$ for the solution started in x and use the decomposition where solves the stochastic heat equation

(2.1)

The remainder term v solves

(2.2)

where are the 2nd and 3rd Wick powers of the solution to the stochastic heat equation . The random distributions and can be constructed as limits of and , where is a spatial Galerkin approximation of , and $\mathfrak {R}_N$ is a renormalisation constant which diverges logarithmically in the regularisation parameter N. The value of $\mathfrak {R}_N$ is given by

(2.3)

Note that is stationary in the space variable z, hence the expectation is independent of z. We refer the reader to [6, Lemma 3.2], [24, Section 2] for more details on the construction of the Wick powers. We recall that and can be realised as continuous (in time) processes taking values in $\mathcal {C}^{-\alpha }$ for $\alpha >0$ and that $\mathbb {P}$-almost surely for every $T>0$, and $\alpha '>0$

(2.4)

The blow-up of and for t close to 0 is due to the fact that we define the stochastic objects and with zero initial condition, but we work with a time-independent renormalisation constant $\mathfrak {R}_N$ (see (2.3)). We define the stochastic heat equation with a Laplacian with mass 1 because this allows us to prove exponential moment bounds of and which hold uniformly in time (see Proposition D.1). Throughout the paper we use to refer to all the stochastic objects and simultaneously. In this notation (2.4) turns into

We fix $\alpha _0\in (0,\frac{1}{3})$ (to measure the regularity of the initial condition x in $\mathcal {C}^{-\alpha _0}$), $\beta >0$ (to measure the regularity of v in $\mathcal {C}^\beta$) and $\gamma >0$ (to measure the blow-up of $\Vert v(t;x)\Vert _{\mathcal {C}^\beta }$ for t close to 0) such that

$$\begin{aligned} \gamma<\frac{1}{3}, \quad \frac{\alpha _0+\beta }{2} < \gamma . \end{aligned}$$

(2.5)

We also assume that $\alpha '>0$ and $\alpha >0$ in (2.4) satisfy

$$\begin{aligned} \alpha '<\gamma , \quad \alpha<\alpha _0, \quad \frac{\alpha +\beta }{2} + 2 \gamma < 1. \end{aligned}$$

(2.6)

In [24, Theorems 3.3 and 3.9]) it was shown that for every $x\in \mathcal {C}^{-\alpha _0}$ there exist a unique solution $v\in C\left( (0,\infty );\mathcal {C}^\beta \right)$ of (2.2) such that for every $T>0$

$$\begin{aligned} \sup _{t\le T}(t\wedge 1)^\gamma \Vert v(t;x)\Vert _{\mathcal {C}^\beta }<\infty . \end{aligned}$$

Remark 2.1

In Condition (2.5) $\beta$ has to be strictly less than $\frac{2}{3}$. This is necessary if one wants to treat all of the terms arising in a fixed point problem for (2.2) with the same norm for v. A simple post-processing of [24, Theorems 3.3 and 3.9] shows that in fact v is continuous in time taking values in $\mathcal {C}^{2-\lambda }$ for any $\lambda > \alpha$.

Equations (2.1), (2.2) suggest that indeed X can be seen as a perturbation of the Allen–Cahn equation (1.1), because the terms and in (2.2) all appear with a positive power of $\varepsilon$. It is important to note that v is much more regular than X. The irregular part of $X(\cdot ;x)$ is . Therefore differences of solutions are much more regular than solutions themselves.

We repeatedly work with restarted stochastic terms: we define as the solution of

and let and be its Wick powers. By [24, Proposition 2.3] for every $s>0$, are independent of $\mathcal {F}_s$ and equal in law to . For $t\ge s$ we can define a restarted remainder $v_s(t;X(s;x))$ through the identity . Rearranging (2.2) and using the pathwise identities in [24, Corollary 2.4] one can see that $v_s$ solves

(2.7)

In [24, Theorem 4.2] this is used to prove the Markov property for $X(\cdot ;x)$.

3 Main results

In this article we prove the following main theorem.

Theorem 3.1

For every $\kappa >0$ there exist $\delta _0, a_0, C>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&\inf _{\Vert x - (\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0}\\&\qquad \mathbb {P}\left( \sup _{\Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0} \frac{\Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta }}{\Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}} \le C \mathrm {e}^{-(2-\kappa )t} \text { for every } t\ge 1\right) \\&\quad \ge 1 - \mathrm {e}^{-a_0/\varepsilon }. \end{aligned}$$

Proof

See Sect. 4.2. $\square$

This theorem is a variant of [17, Corollary 3.1] in space dimension $d=2$, but in that work the supremum is taken over both x and y inside the probability measure. We also obtain this version of the theorem as a corollary.

Corollary 3.2

For every $\kappa >0$ there exist $\delta _0,a_0,C>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&\mathbb {P}\left( \sup _{x,y\in {\bar{B}}_{\mathcal {C}^{-\alpha _0}}(\pm 1;\delta _0)} \frac{\Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta }}{\Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}} \le C \mathrm {e}^{-(2-\kappa )t} \text { for every } t\ge 1\right) \\&\quad \ge 1 - \mathrm {e}^{-a_0/\varepsilon }. \end{aligned}$$

Proof

See Sect. 4.2. $\square$

Remark 3.3

The restriction $t\ge 1$ in Theorem 3.1 appears only because we measure $y-x$ in a lower regularity norm than $X(t;y) - X(t;x)$. To prove the theorem we first prove Theorem 4.9 were we assume that $y-x\in \mathcal {C}^\beta$ and in this case we prove a bound which holds for every $t>0$.

Remark 3.4

Theorem 3.1 is an asymptotic coupling of solutions that start close to the same minimiser. In [17, Proposition 3.4] it was shown that in the 1-dimensional case, solutions which start with initial conditions x and y close to different minimisers also contract exponentially fast, but only after time $T_\varepsilon \propto \mathrm {e}^{[(V(0)-V(\pm 1))+\eta ]/\varepsilon }$ for any $\eta >0$. This is the “typical” time needed for one of the two profiles to jump close to the other minimiser. We expect that Theorem 3.1 and the large deviation theory developed in [14] could be combined to prove a similar result in the case $d=2$.

As an application of Theorem 3.1 we prove an Eyring–Kramers law for the transition times of X. Before we state our main result in this direction let us briefly introduce some extra notation.

For $\delta \in (0,1/2)$ and $\alpha >0$ we define the symmetric subsets $A(\alpha ;\delta )$ and $B(\alpha ;\delta )$ of $\mathcal {C}^{-\alpha }$ by

$$\begin{aligned} A(\alpha ;\delta ):= & {} \left\{ f\in \mathcal {C}^{-\alpha } : {\bar{f}} \in [-1-\delta ,-1+\delta ], \, f - {\bar{f}} \in D_{\perp } \right\} \end{aligned}$$

(3.1)

$$\begin{aligned} B(\alpha ;\delta ):= & {} \left\{ f\in \mathcal {C}^{-\alpha } : {\bar{f}} \in [1-\delta ,1+\delta ], \, f - {\bar{f}} \in D_{\perp }\right\} \end{aligned}$$

(3.2)

where $D_\perp$ is the closed ball of radius $\delta$ in $\mathcal {C}^{-\alpha }$ and ${\bar{f}} = L^{-2} \langle f, 1 \rangle$. For $x\in A(\alpha ;\delta )$ we define the transition time

$$\begin{aligned} \tau _{B(\alpha ;\delta )}(X(\cdot ;x)) := \inf \left\{ t>0: X(t;x)\in B(\alpha ;\delta )\right\} . \end{aligned}$$

Last, for $k\in \mathbb {Z}^2$ we let

$$\begin{aligned} \lambda _k := \left( \frac{2\pi |k|}{L}\right) ^2 -1 \quad \text {and} \quad \nu _k : = \left( \frac{2\pi |k|}{L}\right) ^2 +2 = \lambda _k + 3. \end{aligned}$$

(3.3)

The sequences $\{\lambda _k\}_{k\in \mathbb {Z}^2}$ and $\{\nu _k\}_{k\in \mathbb {Z}^2}$ are the eigenvalues of the operators $-\Delta -1$ and $-\Delta + 2$ endowed with periodic boundary conditions.

With this notation at hand the Eyring–Kramers law can be expressed as follows. Notice that by symmetry the same result holds if we swap the neighbourhoods of $-1$ and 1 below.

Theorem 3.5

There exist $\delta _0>0$ such that the following holds. For every $\alpha \in (0,\alpha _0)$ and $\delta \in (0,\delta _0)$ there exist $c_+,c_->0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&\sup _{x\in {A(\alpha _0;\delta )}} \mathbb {E}\tau _{B(\alpha ;\delta )} (X(\cdot ;x)) \nonumber \\&\qquad \le \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} } \mathrm {e}^{\left( V(0) - V(-1)\right) /\varepsilon }\left( 1+c_+\sqrt{\varepsilon }\right) \nonumber \\&\inf _{x\in A(\alpha _0;\delta )} \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \nonumber \\&\qquad \ge \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} } \mathrm {e}^{\left( V(0) - V(-1)\right) /\varepsilon }\left( 1-c_-\varepsilon \right) . \end{aligned}$$

(3.4)

Proof

See Sect. 7.3. $\square$

4 Proof of the exponential loss of memory

In this section we prove the exponential loss of memory. In Sect. 4.1 we present the basic ingredients needed in the proof and in Sect. 4.2 we give the proofs of Theorem 3.1 and Corollary 3.2.

4.1 Methodology

We define two sequences $\{\nu _i(x)\}_{i\ge 1}$ and $\{\rho _i(x)\}_{i\ge 1}$ of stopping times which partition our time axis and allow us to keep track of the time spent close to and away from the minimisers $\pm 1$ (see Fig. 1 for a sketch). On the “good” intervals $[\rho _{i-1}(x), \nu _i(x)]$ we require both the restarted diagrams to be small and the restarted remainder $v_{\rho _{i-1}(x)}$ to be close to $\pm 1$. The “bad” intervals $[\nu _i(x),\rho _i(x)]$ end when $X(\cdot ;x)$ re-enters a small neighbourhood of the minimisers. The stopping times $\rho _i(x)$ are defined in terms of the $\mathcal {C}^{-\alpha _0}$ norm for $X(\cdot ;x)$, while we define good intervals in terms of the stronger $\mathcal {C}^\beta$ topology for $v_{\rho _{i-1}(x)}$. To connect the two, we need to allow for a blow-up close to the starting point of the “good” intervals.

Fig. 1

A partition of the time axis with respect to the times $\nu _i(x)$ and $\rho _i(x)$. The “good” intervals are “typically” much larger than the “bad” intervals

Definition 4.1

For $x\in \mathcal {C}^{-\alpha _0}$ we define the sequence of stopping times $\{\rho _i(x)\}_{i\ge 0}, \{\nu _i(x)\}_{i\ge 1}$ recursively by $\rho _0(x) = 0$ and

We now define the time increments

$$\begin{aligned}&\tau _i(x) = \nu _i(x) - \rho _{i-1}(x) \nonumber \\&\sigma _i(x) = \rho _i(x) - \nu _i(x). \end{aligned}$$

(4.1)

The process $X( \cdot ; x)$ is expected to spend long time intervals close to the minimisers $\pm 1$, which corresponds to large values of $\tau _i(x)$. Large values of $\sigma _i(x)$ are “atypical”. This behaviour is established Propositions 6.3 and 6.6.

The following proposition shows contraction on the “good” intervals. We distinguish between the cases (4.2) and (4.3) for $y-x$ that lie in $\mathcal {C}^\beta$ and $\mathcal {C}^{-\alpha _0}$ respectively. The Da Prato–Debussche decomposition shows that differences of any two profiles lie in $\mathcal {C}^\beta$ for any $t>0$ but at $t=0$ they maintain the irregularity of the initial conditions. Hence we only use (4.3) on the first “good” interval.

Proposition 4.2

For every $\kappa >0$ there exist $\delta _0,\delta _1,\delta _2>0$ and $C>0$ such that if $\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0$ and $y-x\in \mathcal {C}^\beta , \Vert y-x\Vert _{\mathcal {C}^\beta } \le \delta _0$ then

$$\begin{aligned} \Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta } \le C\exp \left\{ -\left( 2-\frac{\kappa }{2}\right) t\right\} \Vert y-x\Vert _{\mathcal {C}^\beta }, \end{aligned}$$

(4.2)

for every $t\le \tau _1(x)$ defined with respect to $\delta _1$ and $\delta _2$. If we only assume that $\Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0$ then

$$\begin{aligned} (t\wedge 1)^\gamma \Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta } \le C \exp \left\{ -\left( 2-\frac{\kappa }{2}\right) t\right\} \Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}, \end{aligned}$$

(4.3)

for every $t\le \tau _1(x)$.

Proof

See Sect. 5.1. $\square$

Our next aim is to control the growth of the differences on the “bad” intervals in terms of the stochastic objects . This is done by partitioning the intervals $[\nu _i(x), \rho _i(x)]$ into tiles of length one. To achieve independence we restart the stochastic objects at the starting point of each tile.

Definition 4.3

For $k\ge 0$ and $\rho \ge \nu \ge 0$ let $t_k = \nu +k$. For $k \ge 1$ we define a random variable $L_k(\nu ,\rho )$ by

(4.4)

In our analysis we use a second tiling defined by setting $s_k= t_k + \frac{1}{2}$, i.e. the tiles $[t_k,t_{k+1}]$ and $[s_k, s_{k+1}]$ overlap. In order to bound $X(t;y) - X(t;x)$ on a time interval $[t_k,s_k]$ we restart the stochastic objects at $s_{k-1}$ and write $X(t;y) - X(t;x) = v_{s_{k-1}}(t;X(s_{k-1};y)) - v_{s_{k-1}}(t;X(s_{k-1};x))$. In Lemma 5.1 we upgrade the strong a priori bound obtained in [24, Proposition 3.7] to get a control on the $\mathcal {C}^\beta$ norm of both remainders. This bound holds uniformly over all possible values of $X(s_{k-1};y)$ and $X(s_{k-1};x)$ and while the bound allows for a blow-up for times t close to $s_{k-1}$ it holds uniformly over all times in $[t_k,s_k]$. Ultimately, the bound only depends on $L_k(\nu +\frac{1}{2},\rho )$ in a polynomial way as shown in Fig. 2. Then we can use the local Lipschitz property of the non-linearity in (2.2) to bound the exponential growth rate of $X(t;y) - X(t;x)$. For the first interval $[t_0,t_1]$ we do not use this trick, because we want to avoid bounds that depend on the realisation of the white noise outside of $[\nu , \rho ]$. On this interval, we make use of an a priori assumption that we have some control on $\Vert X(\nu ;y)\Vert _{\mathcal {C}^{-\alpha _0}}$ and $\Vert X(\nu ;x)\Vert _{\mathcal {C}^{-\alpha _0}}$.

Fig. 2

Bounds on the $\mathcal {C}^\beta$ norm of the restarted remainder v on the overlapping tiles of the partition of $[\nu ,\rho ]$. On a time interval $[t_k,s_k]$ we restart the stochastic objects at time $s_{k-1}$ and bound $v_{s_{k-1}}$ by a polynomial function of $L_k\left( \nu +\frac{1}{2},\rho \right)$. On a time interval $[s_k,t_{k+1}]$ we restart the stochastic objects at time $t_k$ and bound $v_{t_k}$ by a polynomial function of $L_k\left( \nu ,\rho \right)$

Proposition 4.4

Let $R>0$. Then there exists a constant $C\equiv C(R)>0$ such that for every $\Vert X(\nu ;x)\Vert _{\mathcal {C}^{-\alpha _0}}, \Vert X(\nu ;y)\Vert _{\mathcal {C}^{-\alpha _0}} \le R, \rho >\nu \ge 0$ and $t\in [\nu ,\rho ]$

$$\begin{aligned} \Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta } \le C \exp \left\{ L(\nu ,\rho ;t-\nu )\right\} \Vert X(\nu ;y) - X(\nu ;x)\Vert _{\mathcal {C}^\beta }, \end{aligned}$$

(4.5)

where

$$\begin{aligned} L(\nu ,\rho ;t-\nu ) = \frac{c_0}{2} \sum _{k=1}^{\lfloor t-\nu \rfloor } \sum _{l=0,\frac{1}{2}} \left( 1\vee L_k(\nu +l,\rho )\right) ^{p_0} + L_0 (t-\nu ), \end{aligned}$$

(4.6)

for $L_k$ as in (4.4), and for some constants $p_0\ge 1$ and $c_0\equiv c_0(R),L_0\equiv L_0(R)\ge 0$.

Proof

See Sect. 5.2. $\square$

If we assume that $y-x\in \mathcal {C}^\beta$, combining the estimates in Propositions 4.2 and 4.4 suggest the bound

$$\begin{aligned}&\Vert X(\rho _N(x);y) - X(\rho _N(x);x)\Vert _{\mathcal {C}^\beta } \nonumber \\&\quad \le \exp \left\{ \sum _{i\le N} \left[ -\left( 2-\frac{\kappa }{2}\right) \tau _i + L(\nu _i(x),\rho _i(x);\sigma _i(x))+ 2\log C\right] \right\} \Vert y-x\Vert _{\mathcal {C}^\beta }, \end{aligned}$$

(4.7)

for any $N\ge 1$. If we can show that the exponents satisfy

$$\begin{aligned} \sum _{i\le N}\left[ -\left( 2-\frac{\kappa }{2}\right) \tau _i + L(\nu _i(x),\rho _i(x);\sigma _i(x))+2\log C\right] \le -(2-\kappa )\rho _N(x), \end{aligned}$$

then (4.7) yields exponential contraction at time $\rho _N(x)$ with rate $2-\kappa$. The difference of the right hand side and the left hand side of the last inequality is given by the random walk $S_N(x)$ in the next definition.

Definition 4.5

Let $\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0$. We define the random walk $(S_N(x))_{N\ge 1}$ by

$$\begin{aligned} S_N(x) := \sum _{i\le N} \left[ \frac{\kappa }{2} \tau _i(x) - \big (L(\nu _i(x),\rho _i(x);\sigma _i(x)) + (2-\kappa ) \sigma _i(x) + M_0\big )\right] , \end{aligned}$$

where $M_0 = 2\log C$ for $C>0$ as in Propositions 4.2 and 4.4 .

Fig. 3

“Typical” realisations of a random walk $S_N = \sum _{i\le N} (f_i - g_i)$ for $f_i \sim \mathrm {e}^{0.5/\varepsilon }\exp (1), g_i \sim \mathrm {e}^{0.1/\varepsilon } \mathrm {Weibull}(0.5,1), N=50$ and $\varepsilon =0.01$. The choice of a Weibull distribution here captures the fact that the random variables $L(\nu _i(x),\rho _i(x);\sigma _i(x)) + (2-\kappa ) \sigma _i(x) + M_0$ in Definition 4.5 have stretched exponential tails as shown in Proposition 6.7

The next proposition shows that the random walk $S_N(x)$ stays positive for every $N \ge 1$ with overwhelming probability (see Fig. 3 for an illustration). The proof is based on a variant of the classical Cramér–Lundberg model in risk theory (see [8, Chapter 1.2]). In this classical model a random walk $S_N=\sum _{i\le N}(f_i - g_i)$ with i.i.d. exponential random variables $f_i$ and i.i.d. non-negative random variables $g_i$ is considered. The probability for $S_N$ to stay positive for every $N\ge 1$ can be calculated explicitly in terms of the expectations of $f_i$ and $g_i$ using a renewal equation. In our case we use the Markov property and Propositions 6.3 and Proposition 6.7 to compare the random walk $S_N(x)$ in Definition 4.5 to this classical case.

Remark 4.6

If the family $\{L(\nu _i(x),\rho _i(x);\sigma _i(x)) + (2-\kappa ) \sigma _i(x) + M_0\}_{i\ge 1}$ had exponential moments, a simple exponential Chebyshev argument would imply the following proposition without any reference to the Cramér–Lundberg model. However, by (4.4) and (4.6) one sees that $L(\nu _i(x),\rho _i(x);\sigma _i(x))$ is a polynomial of potentially high degree in the explicit stochastic objects (which are themselves polynomials of the Gaussian noise $\xi$). Hence, we cannot expect more than stretched exponential moments, and indeed, such bounds are established in Proposition 6.7. In the proof of the next proposition we also use an exponential Chebyshev argument, but only to compare $\frac{\kappa }{2}\tau _i(x)$ with a suitable exponential random variable which does not depend on x.

Proposition 4.7

For every $\kappa >0$ there exist $a_0>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \inf _{\Vert x - (\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0} \mathbb {P}(S_N(x) \ge 0 \text { for every } N\ge 1) \ge 1-\mathrm {e}^{- a_0/\varepsilon }. \end{aligned}$$

(4.8)

Proof

See Sect. 6.3. $\square$

4.2 Proofs of Theorem 3.1 and Corollary 3.2

We first treat the case where $y-x\in \mathcal {C}^\beta$: let $x\in \mathcal {C}^{-\alpha _0}$ such that $\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0$ and let y be such that $y-x\in \mathcal {C}^\beta$ and $\Vert y-x\Vert _{\mathcal {C}^\beta }\le \delta _0$. We also write $Y(t) = X(t;y) - X(t;x)$. We consider the event

$$\begin{aligned} \mathcal {S}(x) = \left\{ S_N(x) \ge 0 \text { for every } N\ge 1\right\} , \end{aligned}$$

(4.9)

for $S_N(x)$ as in Definition 4.5.

We first prove the following proposition which provides explicit estimates on the differences at the stopping times $\nu _N(x)$ and $\rho _N(x)$ for every $N\ge 1$ and $\omega \in \mathcal {S}(x)$ by iterating Propositions 4.2 and 4.4. To shorten the notation we drop the explicit dependence on the starting point x in the stopping times $\nu _N$ and $\rho _N$ and the random walk $S_N$. We also drop the dependence on the realisation $\omega$ but we assume throughout that $\omega \in \mathcal {S}(x)$.

Proposition 4.8

For any $\kappa >0$ let $C>0$ be as in Proposition 4.2. Then for every $\omega \in \mathcal {S}(x)$ and $N\ge 1$

$$\begin{aligned} \Vert Y(\nu _N)\Vert _{\mathcal {C}^\beta }&\le C \exp \left\{ -S_{N-1}\right\} \exp \left\{ -\frac{\kappa }{2}\tau _N\right\} \exp \left\{ -(2-\kappa ) \nu _N\right\} \Vert Y(0)\Vert _{\mathcal {C}^\beta } \end{aligned}$$

(4.10)

$$\begin{aligned} \Vert Y(\rho _N)\Vert _{\mathcal {C}^\beta }&\le \exp \left\{ -S_N\right\} \exp \left\{ -(2-\kappa )\rho _N\right\} \Vert Y(0)\Vert _{\mathcal {C}^\beta } . \end{aligned}$$

(4.11)

Proof

We prove our claim by induction on $N\ge 1$, observing that it is obvious for $N=0$.

To prove (4.10) for $N+1$ we first notice that by the definition of $\rho _N$ we have that $\Vert X^\varepsilon (\rho _N;x)-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0$ and since $\omega \in \mathcal {S}(x)$ (4.11) implies that $\Vert Y(\rho _N)\Vert _{\mathcal {C}^\beta } \le \delta _0$. Hence we can use (4.2) to get

$$\begin{aligned} \Vert Y(\nu _{N+1})\Vert _{\mathcal {C}^\beta } \lesssim \exp \left\{ -\frac{\kappa }{2}\tau _{N+1}\right\} \exp \left\{ -(2-\kappa )\tau _{N+1}\right\} \Vert Y(\rho _{N})\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

Combining with the estimate on $\Vert Y(\rho _N)\Vert _{\mathcal {C}^\beta }$ the above implies (4.10) for $N+1$.

To prove (4.11) for $N+1$ we first notice that by Proposition 6.2$\Vert X(\nu _{N+1};x)\Vert _{\mathcal {C}^{-\alpha _0}}\le 2\delta _1+1$. This bound, (4.10) for $N+1$ and the triangle inequality imply that $\Vert X(\nu _{N+1};y)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0 + 2\delta _1+1$. Hence we can use Proposition 4.4 for $\nu = \nu _{N+1}, \rho = \rho _{N+1}$ and $R=\delta _0 + 2\delta _1+1$ to obtain

$$\begin{aligned} \Vert Y(\rho _{N+1})\Vert _{\mathcal {C}^\beta } \lesssim \exp \left\{ L(\nu _{N+1},\rho _{N+1};\sigma _{N+1})\right\} \Vert Y(\nu _{N+1})\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

If we combine with (4.10) for $N+1$ we have that

$$\begin{aligned} \Vert Y(\rho _{N+1})\Vert _{\mathcal {C}^\beta }&\le \exp \left\{ L(\nu _{N+1},\rho _{N+1};\sigma _{N+1})+M_0\right\} \exp \left\{ -S_{N}\right\} \\&\quad \times \exp \left\{ -\frac{\kappa }{2}\tau _{N+1}\right\} \\&\quad \exp \left\{ -(2-\kappa ) \nu _{N+1}\right\} \Vert Y(0)\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

We then rearrange the terms to obtain (4.11), which completes the proof. $\square$

We are ready to prove the following version of Theorem 3.1 for sufficiently smooth initial conditions.

Theorem 4.9

For every $\kappa >0$ there exist $\delta _0, a_0, C>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&\inf _{\Vert x - (\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0}\\&\qquad \mathbb {P}\left( \sup _{\begin{array}{c} y-x\in \mathcal {C}^\beta \\ \Vert y-x\Vert _{\mathcal {C}^\beta } \le \delta _0 \end{array}} \frac{\Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta }}{\Vert y-x\Vert _{\mathcal {C}^\beta }} \le C \mathrm {e}^{-(2-\kappa )t} \text { for every } t\ge 0\right) \\&\quad \ge 1 - \mathrm {e}^{-a_0/\varepsilon }. \end{aligned}$$

Proof

Let $\omega \in \mathcal {S}(x)$ as in (4.9). For any $t>0$ there exists $N\equiv N(\omega )\ge 0$ such that $t\in [\rho _N, \nu _{N+1})$ or $t\in [\nu _{N+1}, \rho _{N+1})$.

If $t\in [\rho _N, \nu _{N+1})$ then

$$\begin{aligned}&\Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta } \\&\quad {\mathop {\lesssim }\limits ^{(4.2), (4.11)}} \exp \left\{ -\left( 2-\frac{\kappa }{2}\right) (t - \rho _N)\right\} \Vert X(\rho _N;y)) - X(\rho _N;x)\Vert _{\mathcal {C}^\beta } \\&\quad = \exp \left\{ -\frac{\kappa }{2}(t-\rho _N)\right\} \exp \left\{ -(2-\kappa )(t-\rho _N) \right\} \Vert X(\rho _N;y)) - X(\rho _N;y)\Vert _{\mathcal {C}^\beta } \\&\quad {\mathop {\lesssim }\limits ^{(4.11)}} \exp \left\{ -(2-\kappa ) t \right\} \Vert y - x\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

If $t\in [\nu _{N+1}, \rho _{N+1})$ then

$$\begin{aligned}&\Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta } \\&\quad {\mathop {\lesssim }\limits ^{(4.5)}} \exp \{L(\nu _{N+1},\rho _{N+1};t-\nu _{N+1})\} \Vert X(\nu _{N+1};y) - X(\nu _{N+1};x)\Vert _{\mathcal {C}^\beta } \\&\quad = \exp \{L(\nu _{N+1},\rho _{N+1};t-\nu _{N+1})+(2-\kappa )(t-\nu _{N+1})\} \\&\qquad \times \exp \{-(2-\kappa )(t-\nu _{N+1})\} \Vert X(\nu _{N+1};y) - X(\nu _{N+1};x)\Vert _{\mathcal {C}^\beta } \\&\quad {\mathop {\lesssim }\limits ^{(4.10),\omega \in \mathcal {S}(x)}} \exp \{-(2-\kappa )t\} \Vert y - x\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

By Proposition 4.7 there exist $a_0>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \inf _{\Vert x - (\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0}\mathbb {P}(\mathcal {S}(x)) \ge 1 - \mathrm {e}^{-a_0/\varepsilon }, \end{aligned}$$

which completes the proof. $\square$

We are now ready to prove Theorem 3.1 and Corollary 3.2.

Proof of Theorem 3.1

This is a consequence of (4.3), Proposition 6.3 and Theorem 4.9. Let $\delta _1,\delta _2>0$ sufficiently small such that $\delta _1+\delta _2<\delta _0$ and assume that $\tau _1(x)\ge 1$. By the definition of $\tau _1(x)$

If we also choose $\delta _0'< \delta _0$ by (4.3) we have that for every $\Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0'$

$$\begin{aligned} \Vert X(1;y) - X(1;x)\Vert _{\mathcal {C}^\beta } \lesssim \Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}. \end{aligned}$$

The probability of the event $\{\tau _1(x) \ge 1\}$ can be estimated from below by Proposition 6.3 uniformly in $\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0'$. Combining with Theorem 4.9 completes the proof. $\square$

Proof of Corollary 3.2

We only prove the case where initial conditions are close to the minimiser 1. We fix $\delta _0',\delta _1'>0$ such that $2\delta _0' <\delta _0$ and $\delta _0'+\delta _1'< \delta _1$. By Proposition 6.2 if we chose $\delta _2$ sufficiently small then

• uniformly for $\Vert y-1\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0'$.

This together with (4.3) implies that for every $x,y\in B_{\mathcal {C}^{-\alpha _0}}(1;\delta _0')$

$$\begin{aligned} \Vert X(1;y) - X(1;x)\Vert _{\mathcal {C}^\beta } \lesssim \Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}} \lesssim \delta _0'. \end{aligned}$$

Let

$$\begin{aligned} \omega \in \mathcal {S}':= \left\{ \sup _{\Vert y-1\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0'} \frac{\Vert X(t;y) - X(t;1)\Vert _{\mathcal {C}^\beta }}{\Vert y-1\Vert _{\mathcal {C}^{-\alpha _0}}} \le C \mathrm {e}^{-(2-\kappa )t} \text { for every } t\ge 1\right\} , \end{aligned}$$

$t\ge 1$ and $y\in B_{\mathcal {C}^{-\alpha _0}}(-1;\delta _0')$. Then

• $\sup _{s \le t \le T} (t-s)^\gamma \Vert v_s(t;X(s;1))- (\pm 1)\Vert _{\mathcal {C}^\beta }\le \delta _1' \Rightarrow \sup _{s \le t \le T} (t-s)^\gamma \Vert v_s(t;X(s;y))- (\pm 1)\Vert _{\mathcal {C}^\beta }\le \delta _1$ for $T,s\ge 1$.

• $\Vert X(t;1)-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0' \Rightarrow \Vert X(t;y)-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0$.

This implies that if we consider the process X(t; y) for $t\ge 1$, the times $\nu _i(X(1;y))$ and $\rho _i(X(1;y))$ of Definition 4.1 for $\delta _0,\delta _1$ and $\delta _2$ can be replaced by the times $\nu _i(X(1;1))$ and $\rho _i(X(1;1))$ for $\delta _0',\delta _1'$ and the same $\delta _2$. Hence the corresponding random walk $S_N(X(1;y))$ in Definition 4.5 can be replaced by $S_N(X(1;1))$.

We can now repeat the proof of Theorem 4.9 for the difference $X(t;y) - X(t;x), t\ge 1$, step by step, replacing the event in (4.9) by

(4.12)

This allows us to prove that

$$\begin{aligned}&\Vert X(t;y) - X(t;x)\Vert _{\mathcal {C}^\beta } \le C \mathrm {e}^{-(2-\kappa )(t-1)} \Vert X(1;y) - X(1;x)\Vert _{\mathcal {C}^\beta } \\&\quad \le C \mathrm {e}^{-(2-\kappa )t} \Vert y - x\Vert _{\mathcal {C}^{-\alpha _0}}, \end{aligned}$$

uniformly in $y,x\in B_{\mathcal {C}^{-\alpha _0}}(1;\delta _0')$.

To estimate the event in (4.12) we use Theorem 3.1 and Propositions 6.1 and 4.7. This completes the proof. $\square$

5 Pathwise estimates on the difference of two profiles

In this section we prove Propositions 4.2 and Propositions 4.4. Our analysis here is pathwise and uses no probabilistic tools.

5.1 Proof of Proposition 4.2

Proof of Proposition 4.2

We only prove (4.3). To prove (4.2) we follow the same strategy as below. However in this case we do not need to encounter the blow-up of $\Vert Y(t)\Vert _{\mathcal {C}^\beta }$ close to 0 and hence we omit the proof since it poses no extra difficulties.

Let $Y(t) = X(t;y) - X(t;x)$ and notice that from (2.2) we get

We use the identity $v(\cdot ;y) = v(\cdot ;x) +Y$ to rewrite this equation in the form

where $\mathtt {Error}(v(\cdot ;x);Y) = -Y^3 - 3v(\cdot ;x) Y^2$ collects all the terms which are higher order in Y. Then

(5.1)

We set

$$\begin{aligned} {\tilde{\kappa }} = \sup _{t\le \tau _1(x)} (t\wedge 1)^{2\gamma }\Vert -3\left( v(t;x)^2 - 1\right) \Vert _{\mathcal {C}^\beta }. \end{aligned}$$

Let $\iota = \inf \{t>0: (t\wedge 1)^\gamma \Vert Y(t)\Vert _{\mathcal {C}^\beta } > \zeta \}$ for $1\ge \zeta >\delta _0$ and notice that for $t\le \tau _1(x) \wedge \iota$ using (5.1) we get

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta }&{\mathop {\le }\limits ^{(\mathrm{A.6}), (\mathrm{A.7}),(\mathrm{A.8})}} \mathrm {e}^{-2t} C (t\wedge 1)^{-\frac{\alpha _0+\beta }{2}} \Vert Y(0)\Vert _{\mathcal {C}^{-\alpha _0}}\\&\quad + {\tilde{\kappa }} \int _0^t \mathrm {e}^{-2(t-s)} (s\wedge 1)^{-2\gamma } \Vert Y(s)\Vert _{\mathcal {C}^\beta } \,\mathrm {d}s \\&\quad + \zeta \, C_1 \int _0^t \mathrm {e}^{-2(t-s)} (s\wedge 1)^{-2\gamma } \Vert Y(s)\Vert _{\mathcal {C}^\beta } \,\mathrm {d}s \\&\quad + \delta _2 \, C_2 \int _0^t \mathrm {e}^{-2(t-s)} (t-s)^{-\frac{\alpha +\beta }{2}} (s\wedge 1)^{-\gamma } \Vert Y(s)\Vert _{\mathcal {C}^\beta } \,\mathrm {d}s \\&\quad + \delta _2 \, C_3 \int _0^t \mathrm {e}^{-2(t-s)} (t-s)^{-\frac{\alpha +\beta }{2}} (s\wedge 1)^{-\alpha '} \Vert Y(s)\Vert _{\mathcal {C}^\beta } \,\mathrm {d}s, \end{aligned}$$

where we also use that for $s\le t$

$$\begin{aligned} \Vert \mathtt {Error}(v(s;x);Y(s))\Vert _{\mathcal {C}^\beta } \lesssim \zeta s^{-2\gamma }\Vert Y(s)\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

Choosing $\zeta \le {\tilde{\kappa }}/ C_1$ and $\delta _2 \le \tilde{\kappa }/ C_2\vee C_3$ we have

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta }\le & {} \mathrm {e}^{-2t} C (t\wedge 1)^{-\frac{\alpha +\beta }{2}}\Vert Y(0)\Vert _{\mathcal {C}^{-\alpha _0}} \\&+ {\tilde{\kappa }} \int _0^t \mathrm {e}^{-2(t-s)} (t-s)^{-\frac{\alpha +\beta }{2}} (s\wedge 1)^{-2\gamma } \Vert Y(s)\Vert _{\mathcal {C}^\beta } \,\mathrm {d}s. \end{aligned}$$

Then, for $t\le \tau _1(x) \wedge \iota$, by the generalised Gronwall inequality, Lemma B.1, on $f(t) = (t\wedge 1)^\gamma \Vert Y(t)\Vert _{\mathcal {C}^{\beta }}$ there exist $c>0$ such that

$$\begin{aligned} (t\wedge 1)^\gamma \Vert Y(t)\Vert _{\mathcal {C}^\beta } \le C \exp \Big \{ - 2 t + c {\tilde{\kappa }}^{\frac{1}{1-\frac{\alpha +\beta }{2}-3\gamma }} t + M\Big \} \Vert Y(0)\Vert _{\mathcal {C}^{-\alpha _0}}. \end{aligned}$$

We now fix $\delta _1>0$ such that $c \tilde{\kappa }^{\frac{1}{1-\frac{\alpha +\beta }{2}-3\gamma }} \le \frac{\kappa }{2}$. This implies that for $t\le \tau _1(x) \wedge \iota$

$$\begin{aligned} (t\wedge 1)^\gamma \Vert Y(t)\Vert _{\mathcal {C}^\beta } \le C \exp \left\{ - \left( 2 - \frac{\kappa }{2}\right) t\right\} \Vert Y(0)\Vert _{\mathcal {C}^{-\alpha _0}}. \end{aligned}$$

Finally choosing $\delta _0$ sufficiently small we furthermore notice that $\tau _1(x) \wedge \iota = \tau _1(x)$, which completes the proof of (4.3). $\square$

5.2 Proof of Proposition 4.4

Before we proceed to the proof of Proposition 4.4 we need the following lemma which upgrades the a priori estimates in [24, Proposition 3.7]. Here and below we let $S(t) = \mathrm {e}^{\Delta t}$.

Lemma 5.1

There exist $\alpha ,\gamma ',C>0$ and $p_0\ge 1$ such that if then

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}}\sup _{t\le 1} t^{\gamma '} \Vert v(t;x)\Vert _{\mathcal {C}^\beta } \le C (1\vee L)^{p_0}. \end{aligned}$$

Proof

Throughout this proof we simply write v(t) to denote v(t; x). We first need bounds on $\Vert v(t)\Vert _{L^p}$, for p sufficiently large, and $\int _s^t \Vert \nabla v(r)\Vert _{L^2}^2 \,\mathrm {d}r$. These bounds can be obtained by classical energy estimates. A bound on $\Vert v(t)\Vert _{L^p}$ has already been obtained in [24, Proposition 3.7], which states that for every $p\ge 2$ even

(5.2)

for some exponents $p_n\ge 1$. To bound $\int _s^t \Vert \nabla v(r)\Vert _{L^2}^2 \,\mathrm {d}r$ we need to slightly modify the strategy used in the proof of [24, Proposition 3.7]. In particular, combining [24, Equations (3.13) and (3.22)] and integrating from s to t we obtain the energy inequality,

which implies that

(5.3)

We now upgrade these bounds to bounds on $\Vert v(t)\Vert _{\mathcal {C}^\beta }$. Using the mild form of (2.2) we have for $1\ge t>s>0$

(5.4)

To estimate $\Vert v(t)\Vert _{\mathcal {C}^\beta }$ we use the $L^p$ bound (5.2), the energy inequality (5.3) and the embedding $\mathcal {B}^1_{2,\infty }$ to bound the terms appearing on the right hand side of the last inequality as shown below.

We treat each term in (5.4) separately. Below p may change from term to term and $\alpha ,\lambda$ can be taken arbitrarily small. We write $p_1$ and $p_2$ for conjugate exponents of p, i.e. $\frac{1}{p} = \frac{1}{p_1}+\frac{1}{p_2}$. We also denote by $(1\vee L)^{p_0}$ a polynomial of degree $p_0\ge 1$ in the variable $1\vee L$ where the value of $p_0$ may change from line to line.

Term ${I_1}$:

$$\begin{aligned} I_1 {\mathop {\lesssim }\limits ^{(\mathrm{A.5}), (\mathrm{A.6})}} (t-s)^{-\frac{\beta +\frac{2}{p}}{2}} \Vert v(s)\Vert _{L^p} {\mathop {\lesssim }\limits ^{(5.2)}} (t-s)^{-\frac{\beta +\frac{2}{p}}{2}} s^{-\frac{1}{2}} (1\vee L)^{p_0}. \end{aligned}$$

Term ${I_2}$:

$$\begin{aligned} I_2&{\mathop {\lesssim }\limits ^{(\mathrm{A.5}), (\mathrm{A.6})}} \int _s^t (t-r)^{-\frac{\beta +\frac{2}{p}}{2}} \Vert v(r)^3\Vert _{L^p} \,\mathrm {d}r {\mathop {\lesssim }\limits ^{(5.2)}} (1\vee L)^{p_0} \int _s^t (t-r)^{-\frac{\beta +\frac{2}{p}}{2}} r^{-\frac{3}{2}} \,\mathrm {d}r \\&\qquad \lesssim (1\vee L)^{p_0} s^{-\frac{3}{2}} \int _s^t (t-r)^{-\frac{\beta +\frac{2}{p}}{2}} \,\mathrm {d}r. \end{aligned}$$

Term ${I_3}$:

Term ${I_4}$:

Term ${I_5}$:

Term ${I_6}$:

Term ${I_7}$:

$$\begin{aligned} I_7&{\mathop {\lesssim }\limits ^{(\mathrm{A.5}), (\mathrm{A.6})}} \int _s^t (t-r)^{-\frac{\beta + \frac{2}{p}}{2}} \Vert v(r)\Vert _{L^p} \,\mathrm {d}r {\mathop {\lesssim }\limits ^{(5.2)}} (1\vee L)^{p_0} \int _s^t (t-r)^{-\frac{\beta + \frac{2}{p}}{2}} r^{-\frac{1}{2}} \,\mathrm {d}r \\&\qquad \lesssim (1\vee L)^{p_0} s^{-\frac{1}{2}} \int _s^t (t-r)^{-\frac{\beta + \frac{2}{p}}{2}} \,\mathrm {d}r. \end{aligned}$$

Using Proposition A.9, (5.2) and (5.3) we notice that

$$\begin{aligned} \left( \int _s^t \Vert v(r)\Vert _{\mathcal {B}^1_{2,\infty }}^2 \,\mathrm {d}r\right) ^{\frac{1}{2}}\lesssim & {} \left( \int _s^t \Vert \nabla v(r)\Vert _{L^2}^2 \,\mathrm {d}r\right) ^{\frac{1}{2}} + \left( \int _s^t \Vert v(r)\Vert _{L^2}^2 \,\mathrm {d}r\right) ^{\frac{1}{2}}\\\lesssim & {} (1\vee L)^{p_0} s^{-\frac{1}{2}} . \end{aligned}$$

Combining the above and choosing $s=t/2$ we find $\gamma ' > 0$ such that

$$\begin{aligned} t^{\gamma '}\Vert v(t)\Vert _{\mathcal {C}^\beta } \lesssim (1\vee L)^{p_0} \end{aligned}$$

which completes the proof. $\square$

Proof of Proposition 4.4

We denote by $(1\vee L)^{p_0}$ a polynomial of degree $p_0\ge 1$ in the variable $1\vee L$ where the value of $p_0$ may change from line to line.

For $k\ge 0$ recall that $t_k = \nu + k$ and $s_k = t_k + \frac{1}{2}$. As before, we write $Y(t) = X(t;y) - X(t;x)$.

Let $t\in (t_k,s_k], k\ge 1$. We restart the stochastic terms at time $s_{k-1}$ and write $Y(t) = v_{s_{k-1}}(t;{\tilde{y}}) - v_{s_{k-1}}(t;{\tilde{x}})$ where for simplicity ${\tilde{y}} = X(s_{k-1};y)$ and ${\tilde{x}} = X(s_{k-1};x)$. Together with (2.7), this implies that

Using the mild form of the above equation, now starting at $t_k = s_{k-1}+\frac{1}{2}$, we get

By Lemma 5.1 there exist $\gamma '>0$ such that

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}}\sup _{t\in [s_{k-1}, s_k]} \left( t- s_{k-1}\right) ^{\gamma '} \Vert v_{s_{k-1}}(t;x)\Vert _{\mathcal {C}^\beta } \lesssim \left( 1\vee L_k\left( \nu +\frac{1}{2},\rho \right) \right) ^{p_0}. \end{aligned}$$

Combining the above we get

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta } \lesssim \Vert Y(t_k)\Vert _{\mathcal {C}^\beta } + \left( 1\vee L_k\left( \nu +\frac{1}{2},\rho \right) \right) ^{p_0} \int _{t_k}^t (t- r)^{-\frac{\alpha +\beta }{2}} \Vert Y(r)\Vert _{\mathcal {C}^\beta } \,\mathrm {d}r. \end{aligned}$$

By the generalised Gronwall inequality, Lemma B.1, there exists $c_0>0$ such that

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta } \lesssim \exp \left\{ c_0 \left( 1\vee L_k\left( \nu +\frac{1}{2},\rho \right) \right) ^{p_0} (t - s) \right\} \Vert Y(t_k)\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

(5.5)

Following the same strategy we prove that for $t\in [s_k,t_{k+1}], k\ge 1$,

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta } \lesssim \exp \left\{ c_0 \left( 1\vee L_{k+1}\left( \nu ,\rho \right) \right) ^{p_0} (t - s)\right\} \Vert Y(s_k)\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

(5.6)

Finally, we also need a bound for $t\in [t_0,t_1]$. To obtain an estimate which does not depend on any information before time $t_0$ we use local solution theory. By [24, Theorem 3.3] there exists $t_*\in (t_0,t_1)$ such that

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R}\sup _{r\in [t_0,t_*]} (r-t_0)^\gamma \Vert v_{t_0}(r;x)\Vert _{\mathcal {C}^\beta } \le 1 \end{aligned}$$

and furthermore we can take

$$\begin{aligned} t_* = \left( \frac{1}{C(R\vee L_1(\nu ,\rho ))}\right) ^{p_0}. \end{aligned}$$

By Lemma 5.1 we also have that

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \sup _{r\in \left( t_0,t_1\right] } (r-t_0)^{\gamma '} \Vert v_{t_0}(r;x)\Vert _{\mathcal {C}^\beta } \lesssim (1\vee L_1(\nu ,\rho ))^{p_0}. \end{aligned}$$

Combining these two bounds we get

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R}\sup _{r\in [t_0,t_1]} (r-t_0)^\gamma \Vert v_{t_0}(r;x)\Vert _{\mathcal {C}^\beta } \lesssim (1\vee L_1(\nu ,\rho ))^{p_0}, \end{aligned}$$

(5.7)

where the implicit constant depends on R. Note that $\gamma <\frac{1}{3}$ whereas $\gamma '$ is much larger. We write $Y(t) = v_{t_0}(t;y)-v_{t_0}(t;x)$ and use the mild form starting at $t_0$. We then use (5.7) to bound $\Vert v_{t_0}(t;\cdot )\Vert _{\mathcal {C}^\beta }$ on $[t_0,t_1]$ which implies the estimate

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta } \lesssim \Vert Y(t_0)\Vert _{\mathcal {C}^\beta } + \left( 1\vee L_1\left( \nu ,\rho \right) \right) ^{p_0} \int _{t_0}^t (t- r)^{-\frac{\alpha +\beta }{2}} (r-t_0)^{-2\gamma } \Vert Y(r)\Vert _{\mathcal {C}^\beta } \,\mathrm {d}r. \end{aligned}$$

The extra term $(r-t_0)^{-2\gamma }$ in the last inequality appears because of the blow-up of $v_{t_0}(t;\cdot )$ and for t close to $t_0$. By the generalised Gronwall inequality, Lemma B.1, we obtain that

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta } \lesssim \exp \left\{ c_0 \left( 1\vee L_1\left( \nu ,\rho \right) \right) ^{p_0} (t - s)\right\} \Vert Y(s)\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

(5.8)

For arbitrary $t\in [\nu ,\rho ]$ we glue together (5.5), (5.6) and (5.8) to get

$$\begin{aligned} \Vert Y(t)\Vert _{\mathcal {C}^\beta } \lesssim \exp \left\{ \frac{c_0}{2} \sum _{k=1}^{\lfloor t-\nu \rfloor } \sum _{l=0,\frac{1}{2}} \left( 1\vee L_k(\nu +l,\rho \right) )^{p_0} + L_0 (t-\nu ) \right\} \Vert Y(\nu )\Vert _{\mathcal {C}^\beta }, \end{aligned}$$

for some $L_0>0$ which collects the implicit constants in the inequalities. $\square$

6 Random walk estimates

In this section we prove Proposition 4.7 based mainly on probabilistic arguments. In Sects. 6.1 and 6.2 we provide estimates on $\frac{\kappa }{2} \tau _i(x)$ and $L(\nu _i(x),\rho _i(x);\sigma _i(x)) + (2-\kappa ) \sigma _i(x) + M_0$ from Definition 4.5. In Sect. 6.3 we use these estimates to prove Proposition 4.7.

6.1 Estimates on the exit times

Proposition 6.1

Let $\delta >0$ and . Then there exist $a_0>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \mathbb {P}\left( \tau _{\mathrm {tree}}\le \mathrm {e}^{3a_0/\varepsilon }\right) \le \mathrm {e}^{-3a_0/\varepsilon }. \end{aligned}$$

Proof

First notice that for $N\ge 1$

By Proposition D.1 and the exponential Chebyshev inequality there exists $a_0>0$ such that for every $k\ge 0$

Hence

$$\begin{aligned} \mathbb {P}(\tau _{\mathrm {tree}}\le N) \le N \mathrm {e}^{-6a_0/\varepsilon } \end{aligned}$$

and choosing $N = \mathrm {e}^{3a_0/\varepsilon }$ completes the proof. $\square$

Proposition 6.2

For $\delta _1>0$ sufficiently small there exist $\delta _0,\delta _2>0$ such that if

(6.1)

then for every $\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0$

$$\begin{aligned} \sup _{t\le T}(t\wedge 1)^\gamma \Vert v(t;x) - (\pm 1)\Vert _{\mathcal {C}^\beta } < \delta _1 \end{aligned}$$

and

$$\begin{aligned} \sup _{t\le T} \Vert X(t;x)-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le 2\delta _1. \end{aligned}$$

Proof

Let $u(t) = v(t;x) - (\pm 1)$. By an exact expansion of $-v^3+v$ around $\pm 1$ we have that

(6.2)

where $\mathtt {Error}(u) = -u^3\pm 3u^2$ and $\Vert \mathtt {Error}(u)\Vert _{\mathcal {C}^\beta } \lesssim \Vert u\Vert _{\mathcal {C}^\beta }^3+\Vert u\Vert _{\mathcal {C}^\beta }^2$. Let $T>0$ and $\iota = \inf \{t>0: (t\wedge 1)^\gamma \Vert u(t)\Vert _{\mathcal {C}^\beta } \ge \delta _1\}$ for some $\delta _1>0$ which we fix below. Using the mild form of (6.2) we get

If we furthermore assume (6.1) for $t\le T\wedge \iota$ we obtain that

$$\begin{aligned}&(t\wedge 1)^\gamma \Vert u(t)\Vert _{\mathcal {C}^\beta } \\&\quad \lesssim \delta _0 \mathrm {e}^{-2t} + \delta _1^3 \int _0^t \mathrm {e}^{-2(t-s)} (s\wedge 1)^{-3\gamma } \,\mathrm {d}s + \delta _1^2 \int _0^t \mathrm {e}^{-2(t-s)} (s\wedge 1)^{-2\gamma } \,\mathrm {d}s \\&\qquad + \delta _2 \int _0^t \mathrm {e}^{-2(t-s)} (t-s)^{-\frac{\alpha +\beta }{2}} \left( (s\wedge 1)^{-2\gamma } + (s\wedge 1)^{-\gamma } (s\wedge 1)^{-\alpha '}\right. \\&\left. \qquad + (s\wedge 1)^{-2\alpha '} +1\right) \,\mathrm {d}s. \end{aligned}$$

Then Lemma B.2 implies the bound

$$\begin{aligned} \sup _{t\le T\wedge \iota } (t\wedge 1)^\gamma \Vert u(t)\Vert _{\mathcal {C}^\beta } \lesssim \delta _0 + \delta _1^3 + \delta _1^2 + \delta _2. \end{aligned}$$

Choosing $\delta _0< \frac{\delta _1}{4C}, \delta _1<\frac{1}{4C}$ and $\delta _2<\frac{\delta _1}{4C}$ this implies that $\sup _{t\le T\wedge \iota } (t\wedge 1)^\gamma \Vert u(t)\Vert _{\mathcal {C}^\beta } < \delta _1$ which in turn implies that $\iota \le T$ and proves the first bound.

To prove the second bound we notice that for every $t\le T$

Hence it suffices to prove that $\sup _{t\le T} \Vert u(t)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _1$. Using again the mild form of (6.2) we get

for every $t\le T$. Plugging in (6.1) and the bound $\sup _{t\le T} (t\wedge 1)^\gamma \Vert u(t)\Vert _{\mathcal {C}^\beta }\le \delta _1$ the last inequality implies

$$\begin{aligned} \Vert u(t)\Vert _{\mathcal {C}^\beta }&\lesssim \delta _0 \mathrm {e}^{-2t} + \delta _1^3 \int _0^t \mathrm {e}^{-2(t-s)} (s\wedge 1)^{-3\gamma } \,\mathrm {d}s + \delta _1^2 \int _0^t \mathrm {e}^{-2(t-s)} (s\wedge 1)^{-2\gamma } \,\mathrm {d}s \\&\quad + \delta _2 \int _0^t \mathrm {e}^{-2(t-s)} \left( (s\wedge 1)^{-2\gamma } + (s\wedge 1)^{-\gamma } (s\wedge 1)^{-\alpha '} \right. \\&\left. \quad + (s\wedge 1)^{-2\alpha '} +1\right) \,\mathrm {d}s. \end{aligned}$$

Using again Lemma B.2 we obtain that $\sup _{t\le T}\Vert u(t)\Vert _{\mathcal {C}^{-\alpha _0}} < \delta _1$, which completes the proof.

Proposition 6.3

For every $\kappa >0$ and $\delta _1>0$ sufficiently small there exist $a_0,\delta _0,\delta _2>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0} \mathbb {P}\left( \frac{\kappa }{2}\tau _1(x) \le \mathrm {e}^{2a_0/\varepsilon }\right) \le \mathrm {e}^{-3a_0/\varepsilon }, \end{aligned}$$

where $\tau _1(x)$ is given by (4.1).

Proof

We first notice that there exists $\varepsilon _0>0$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \mathbb {P}\left( \frac{\kappa }{2}\tau _1(x) \le \mathrm {e}^{2a_0/\varepsilon }\right) \le \mathbb {P}\left( \tau _1(x) \le \mathrm {e}^{3a_0/\varepsilon }\right) . \end{aligned}$$

The last probability can be estimated by Propositions 6.2 and 6.1 for $\delta =\delta _2$. $\square$

6.2 Estimates on the entry times

In this section we use large deviation theory and in particular a lower bound of the form

$$\begin{aligned}&\liminf _{\varepsilon \searrow 0}\log \varepsilon \inf _{x\in \aleph }\mathbb {P}(X(\cdot ;x)\in \mathcal {A}(T;x)) \nonumber \\&\quad \ge -\sup _{x\in \aleph } \inf _{\begin{array}{c} f\in \mathcal {A}(T;x) \\ f(0) = x \end{array}} \left\{ \underbrace{\frac{1}{4}\int _0^T \Vert (\partial _t - \Delta )f(t) + f(t)^3 - f(t)\Vert _{L^2}^2 \,\mathrm {d}t}_{=:I(f)}\right\} , \end{aligned}$$

(6.3)

where $\aleph$ is a compact subset of $\mathcal {C}^{-\alpha }$ and $\mathcal {A}(T;x)\subset \{f:(0,T) \rightarrow \mathcal {C}^{-\alpha }\}$ is open. This bound is an immediate consequence of [14] and the remark that the solution map

is jointly continuous on compact time intervals. This estimate implies a “nice” lower bound for the probabilities $\mathbb {P}(X(\cdot ;x)\in \mathcal {A}(T;x))$ if a suitable path $f\in \mathcal {A}(T;x)$ is chosen.

In the next proposition we use the lower bound (6.3) for suitable sets $\aleph$ and $\mathcal {A}(T;x)$ to estimate probabilities of the entry time of X in a neighbourhood of $\pm 1$. In particular, we construct a path $f(\cdot ;x)$ and obtain bounds on $I(f(\cdot ;x))$ uniformly in $x\in \aleph$. This construction is similar in spirit to the one used in [9, proof of Theorem 9.1] for the 1-dimensional analogue of (1.4), although here we consider a slightly different event and the initial conditions are not regular functions since they lie in $\mathcal {C}^{-\alpha _0}$. For this reason we make use of the smoothing properties of the deterministic flow given by Proposition C.3.

Proposition 6.4

Let $\delta _0>0$ and $\sigma (x) = \inf \big \{t > 0 : \min _{x_*\in \{-1,1\}} \Vert X(t;x) - x_*\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0 \big \}$. For every $R,b>0$ there exists $T_0>0$ such that

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \mathbb {P}(\sigma (x) \ge T_0) \le 1 - \mathrm {e}^{-b/\varepsilon }. \end{aligned}$$

Proof

First notice that

$$\begin{aligned} \mathbb {P}(\sigma (x) \le T_0) = \mathbb {P}(\underbrace{\Vert X(T_*;x) - (\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} < \delta _0 \text { for some } T_*\le T_0}_{=:\mathcal {A}(T_0;x)}). \end{aligned}$$

By the large deviation estimate (6.3) it suffices to bound

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \inf _{\begin{array}{c} f \in \mathcal {A}(T_0;x) \\ f(0) = x \end{array}} I(f(\cdot ;x)). \end{aligned}$$

We construct a suitable path $g\in \mathcal {A}(T_0;x)$ and we use the trivial inequality

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \inf _{\begin{array}{c} f \in \mathcal {A}(T_0;x) \\ f(0) = x \end{array}} I(f(\cdot ;x)) \le \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} I(g(\cdot ;x)). \end{aligned}$$

We now give the construction of g which involves five different steps. In Steps 1, 3 and 5, g follows the deterministic flow. The contribution of these steps to the energy functional I is zero. In Steps 2 and 4, g is constructed by linear interpolation. The contribution of these steps is estimated by Lemma 6.5. Below we write $X_{det}(\cdot ;x)$ to denote the solution of (1.1) with initial condition x. We also pass through the space $\mathcal {B}^1_{2,2}$ to use convergence results for $X_{det}(\cdot ;x)$ which hold in this topology (see Propositions C.1 and C.2).

Step 1 (Smoothness of initial condition via the deterministic flow):

Let $\tau _1=1$. For $t\in [0,\tau _1]$ we set $g(t;x) = X_{det}(t;x)$. By Proposition C.3 there exist $C\equiv C(r)>0$ and $\lambda >0$ such that

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \Vert X_{det}(1;x)\Vert _{\mathcal {C}^{2+\lambda }} \le C. \end{aligned}$$

Step 2 (Reach points that lead to a stationary solution):

By Step 1 $g(\tau _1;x)\in B_{\mathcal {C}^{2+\lambda }}(0;C)$ uniformly for $\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R$. Let $\delta >0$ to be fixed below. By compactness there exists $\{y_i\}_{1\le i\le N}$ such that $B_{\mathcal {C}^{2+\lambda }}(0;C)$ is covered by $\cup _{1\le i\le N} B_{\mathcal {B}^1_{2,2}}(y_i;\delta )$. Here we use that $\mathcal {C}^{2+\lambda }$ is compactly embedded in $\mathcal {B}^1_{2,2}$ (see Proposition A.8).

Without loss of generality we assume that $\{y_i\}_{1\le i\le N}$ is such that $y_i\in \mathcal {C}^\infty$ and $X_{det}(t;y_i)$ converges to a stationary solution $-1,0,1$ in $\mathcal {B}^1_{2,2}$. Otherwise we choose $\{y_i^*\}_{1\le i\le N} \in B_{\mathcal {B}^1_{2,2}}(y_i;\delta )$ such that $y_i^* \in \mathcal {C}^\infty$ and relabel them. This is possible because of Proposition C.1.

Let $\tau _2=\tau _1+\tau$, for $\tau >0$ which we fix below. For $t\in [\tau _1,\tau _2]$ we set $g(t;x) = g(\tau _1;x) + \frac{t- \tau _1}{\tau _2-\tau _1} (y_i - g(\tau _1;x))$, where $y_i$ is such that $g(\tau _1;x) \in B_{\mathcal {B}^1_{2,2}}(y_i;\delta )$.

Step 3 (Follow the deterministic flow to reach a stationary solution):

Let $T_i^*$ be such that $X_{det}(t;y_i) \in B_{\mathcal {B}^1_{2,2}}(x_*;\delta )$ for every $t\ge T_i^*$, where $x_*\in \{-1,0,1\}$ is the limit of $X_{det}(t;y_i)$ in $\mathcal {B}^1_{2,2}$, for $\{y_i\}_{1\le i\le N}$ as in Step 2. Let $\tau _3=\tau _2+ \max _{1\le i\le N}T_i^*\vee 1$. For $t\in [\tau _2,\tau _3]$ we set $g(t;x) = X_{det}(t- \tau _2;y_i)$. If $X_{det}(\tau _3-\tau _2;y_i) \in B_{\mathcal {B}^1_{2,2}}(\pm 1;\delta )$ we stop here. Otherwise $X_{det}(\tau _3-\tau _2;y_i)\in B_{\mathcal {B}^1_{2,2}}(0;\delta )\cap B_{\mathcal {C}^{2+\lambda }}(0;C)$ (here we use again Proposition C.3 to ensure that $X_{det}(\tau _3-\tau _2;y_i)\in B_{\mathcal {C}^{2+\lambda }}(0;C)$) and we proceed to Steps 4 and 5.

Step 4 (If an unstable solution is reached move to a point nearby which leads to astable solution):

We choose $y_0\in B_{\mathcal {B}^1_{2,2}}(0;\delta )$ such that $y_0\in \mathcal {C}^\infty$ and $X_{det}(t;y_0)$ converges to either 1 or $-1$ in $\mathcal {B}^1_{2,2}$. This is possible because of Proposition C.2.

Let $\tau _4 = \tau _3 +\tau$ for $\tau >0$ as in Step 2 which we fix below. For $t\in [\tau _3,\tau _4]$ we set $g(t;x) = g(\tau _3;x) + \frac{t - \tau _3}{\tau _4 - \tau _3} (y_0 - g(\tau _3;x))$.

Step 5 (Follow the deterministic flow again to finally reach a stable solution):

Let $T_0^*$ be such that $X_{det}(t;y_0)\in B_{\mathcal {B}^1_{2,2}}(\pm 1;\delta )$ for every $t\ge T_0^*$, where $y_0$ is as in Step 4. Let $\tau _5 = \tau _4+T_0^*\vee 1$. For $t\in [\tau _4,\tau _5]$ we set $g(t;x) = X_{det}(t-\tau _4;y_0)$.

For the path $g(\cdot ;x)$ constructed above we see that after time $t \ge \tau _5, g(t;x)\in B_{\mathcal {B}^1_{2,2}}(\pm 1;\delta )$ for every $\Vert x\Vert -{\mathcal {C}^{-\alpha _0}}\le R$. This implies that $\Vert g(t;x) - (\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} <C\delta$ since by (A.5), $\mathcal {B}^1_{2,2} \subset \mathcal {C}^{-\alpha _0}$. We now choose $\delta >0$ such that $C \delta < \delta _0$ and let $T_0=\tau _5+1$. Then $g\in \mathcal {A}(T_0;x)$.

To bound $I(g(\cdot ;x))$ we split our time interval based on the construction of g i.e. $I_k=[\tau _{k-1},\tau _k]$ for $k=1,\ldots ,4$ and $I_5=[\tau _5,T_0]$. We first notice that for $k=1,3,5$

$$\begin{aligned} \frac{1}{4} \int _{I_k} \Vert (\partial _t-\Delta )g(t;x) +g(t;x)^3 - g(t;x)\Vert _{L^2}^2 \,\mathrm {d}t = 0 \end{aligned}$$

since on these intervals we follow the deterministic flow. For the remaining two intervals, i.e. $k=2,4$, we first notice that by construction $\Vert g(\tau _{k-1};x)\Vert _{\mathcal {C}^{2+\lambda }},\Vert g(\tau _k;x)\Vert _{\mathcal {C}^{2+\lambda }}\le C$. By (A.3), $\mathcal {C}^{2+\lambda } \subset \mathcal {B}^2_{\infty ,2}$ for every $\lambda >0$, hence we also have that $\Vert g(\tau _{k-1};x)\Vert _{\mathcal {B}^2_{\infty ,2}}, \Vert g(\tau _k;x)\Vert _{\mathcal {B}^2_{\infty ,2}}\le C$. We can now choose $\tau$ in Steps 2 and 4 according to Lemma 6.5, which implies that

$$\begin{aligned} \frac{1}{4}\int _{I_k} \Vert (\partial _t - \Delta ) g(t;x) + g(t;x)^3 - g(t;x)\Vert _{L^2}^2 \,\mathrm {d}t \le C \delta . \end{aligned}$$

Hence

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \frac{1}{4}\int _0^{T_0} \Vert (\partial _t - \Delta ) g(t;x) + g(t;x)^3 - g(t;x))\Vert _{L^2}^2 \,\mathrm {d}t \le C\delta . \end{aligned}$$

For $b>0$ we choose $\delta$ even smaller to ensure that $C\delta < b$. Finally, by (6.3) there exists $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \inf _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}} \le R}\mathbb {P}(\sigma (x) \le T_0) \ge \mathrm {e}^{-b/\varepsilon }, \end{aligned}$$

which completes the proof. $\square$

Lemma 6.5

([9, Lemma 9.2]) Let $f(t) = x + \frac{t}{\tau } (y-x)$ such that $\Vert x\Vert _{\mathcal {B}^2_{2,2}}, \Vert y\Vert _{\mathcal {B}^2_{2,2}} \le R$ and $\Vert x-y\Vert _{L^2} \le \delta$. There exist $\tau >0$ and $C\equiv C(R)$ such that

$$\begin{aligned} \frac{1}{4}\int _0^\tau \Vert (\partial _t - \Delta ) f(t) + f(t)^3 - f(t)\Vert _{L^2}^2 \,\mathrm {d}t \le C \delta . \end{aligned}$$

Proof

We first notice that $\partial _t f(t) = \frac{1}{\tau } (y-x)$, hence $\Vert \partial _t f(t)\Vert _{L^2} \le \frac{1}{\tau } \delta$. For the term $\Delta f(t)$ we have

$$\begin{aligned} \Vert \Delta f(t)\Vert _{L^2} \le \Vert \Delta x\Vert _{L^2} +\Vert \Delta y\Vert _{L^2} \lesssim \Vert x\Vert _{\mathcal {B}^2_{2,2}} + \Vert y\Vert _{\mathcal {B}^2_{2,2}} \lesssim R, \end{aligned}$$

where we use that the Besov space $\mathcal {B}^2_{2,2}$ is equivalent with the Sobolev space $H^1$. This is immediate from Definition A.1 for $p=q=2$ if we write $\Vert f*\eta _k\Vert _{L^2}$ using Plancherel’s identity. For the term $f(t)^3 - f(t)$ we have

$$\begin{aligned} \Vert f(t)^3 - f(t)\Vert _{L^2}&\lesssim \Vert f(t)\Vert _{L^6}^3 + \Vert f(t)\Vert _{L^2} {\mathop {\lesssim }\limits ^{(\mathrm{A.4})}} \Vert f(t)\Vert _{\mathcal {B}^0_{6,1}}^3 + \Vert f(t)\Vert _{\mathcal {B}^0_{2,1}} \\&{\mathop {\lesssim }\limits ^{(\mathrm{A.5}),\lambda >0}} \Vert f(t)\Vert _{\mathcal {B}^{\frac{2}{3}+\lambda }_{2,2}}^3 + \Vert f(t)\Vert _{\mathcal {B}^\lambda _{2,2}}\\&{\mathop {\lesssim }\limits ^{(\mathrm{A.2}),\lambda < \frac{1}{3}}} \Vert f(t)\Vert _{\mathcal {B}^2_{2,2}}^3 + \Vert f(t)\Vert _{\mathcal {B}^2_{2,2}}. \end{aligned}$$

Hence for $C\equiv C(R)$

$$\begin{aligned} \frac{1}{2}\int _0^\tau \Vert (\partial _t - \Delta ) f(t) + f(t)^3 - f(t)\Vert _{L^2}^2 \,\mathrm {d}t \le \frac{1}{\tau } \delta ^2 + C \tau . \end{aligned}$$

Choosing $\tau = \delta$ completes the proof. $\square$

In the next proposition we estimate the tails of the entry time of X in a neighbourhood of $\pm 1$ uniformly in the initial condition x. This is achieved by Proposition 6.4 and the Markov property combined with [24, Corollary 3.10] which implies that after time $t=1$ the process $X(\cdot ;x)$ enters a compact subset of the state space with positive probability uniformly in x.

Proposition 6.6

Let $\delta _0>0$ and $\sigma (x) = \inf \big \{t > 0 : \min _{x_*\in \{-1,1\}} \Vert X(t;x) - x_*\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0 \big \}$. For every $b>0$ there exist $T_0>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-{\alpha _0}}} \mathbb {P}(\sigma (x) \ge mT_0) \le \left( 1-\mathrm {e}^{-b/\varepsilon }\right) ^m, \end{aligned}$$

for every $m\ge 1$.

Proof

By [24, Corollary 3.10] we know that $\sup _{x\in \mathcal {C}^{-\alpha _0}} \sup _{t\in (0,1]} t^{\frac{p}{2}} \mathbb {E}\Vert X(t;x)\Vert _{L^p}^p <\infty$, for every $p\ge 2$, and the bound is uniform in $\varepsilon \in (0,1]$ since it only depends polynomially on $\sqrt{\varepsilon }$. Hence by a simple application of Markov’s inequality there exist $R_0>0$ such that

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \sup _{\varepsilon \in (0,1]} \mathbb {P}(\Vert X(1;x)\Vert _{\mathcal {C}^{-\alpha }} > R_0) \le \frac{1}{2}. \end{aligned}$$

(6.4)

By Proposition 6.4 for every $b>0$ there exists $T_0>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R_0} \mathbb {P}(\sigma (x) \ge T_0) \le 1-\mathrm {e}^{-b/\varepsilon }. \end{aligned}$$

(6.5)

Then for every $x\in \mathcal {C}^{-\alpha _0}$ and $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \mathbb {P}(\sigma (x) \ge T_0+1)&\le \mathbb {E}\left( {\mathbf {1}}_{\{\Vert X(1;x)\Vert _{\mathcal {C}^{-\alpha _0}}\le R_0\}} \mathbb {P}(\sigma (X(1;x)) \ge T_0)\right) \nonumber \\&\quad + \mathbb {P}(\Vert X(1;x)\Vert _{\mathcal {C}^{-\alpha _0}} > R_0) \nonumber \\&{\mathop {\le }\limits ^{(6.4), (6.5)}} 1 - \frac{1}{2} \mathrm {e}^{-b/\varepsilon }. \end{aligned}$$

(6.6)

Using the Markov property successively implies for every $m\ge 1$ and $x\in \mathcal {C}^{-\alpha _0}$

$$\begin{aligned} \mathbb {P}(\sigma (x) \ge m (T_0+1)) \le \sup _{y\in \mathcal {C}^{-\alpha _0}}\mathbb {P}(\sigma (y) \ge (T_0+1)) \, \mathbb {P}(\sigma (x) \ge (m-1) (T_0+1)). \end{aligned}$$

(6.7)

Combining (6.6) and (6.7) we obtain that

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {P}(\sigma (x) \ge m (T_0+1)) \le \left( 1 - \frac{1}{2} \mathrm {e}^{-b/\varepsilon }\right) ^m. \end{aligned}$$

The last inequality completes the proof if we relabel b and $T_0$. $\square$

Proposition 6.7

Let $\delta _0>0, \nu _1(x), \rho _1(x)$ as in Definition 4.1, $\sigma _1(x)$ as in (4.1) and $L(\nu _1(x),\rho _1(x);\sigma _1(x))$ as in (4.6). For every $\kappa , M_0,b>0$ there exist $T_0 > 0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&\sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}\left( \left[ L(\nu _1(x),\rho _1(x);\sigma _1(x)) + (2-\kappa ) \sigma _1(x) + M_0\right] ^\frac{1}{p_0} \ge m T_0\right) \\&\quad \le \left( 1-\mathrm {e}^{-b/\varepsilon }\right) ^m, \end{aligned}$$

for every $m\ge 1$ and $p_0\ge 1$ as in (4.6).

Proof

We first condition on $\nu _1(x)$ to obtain the bound

$$\begin{aligned}&\sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}\left( \left[ L(\nu _1(x),\rho _1(x);\sigma _1(x)) + (2-\kappa ) \sigma _1(x) + M_0\right] ^\frac{1}{p_0} \ge m T_0\right) \\&\quad \le \sup _{x\in \mathcal {C}^{-\alpha _0}} \underbrace{\mathbb {P}\left( \left[ L(0,\sigma (x);\sigma (x)) + (2-\kappa ) \sigma (x) + M_0\right] ^\frac{1}{p_0} \ge m T_0\right) }_{=:\mathbb {P}\left( g(\sigma (x))^{\frac{1}{p_0}} \ge m T_0\right) }, \end{aligned}$$

where $\sigma (x) = \inf \left\{ t > 0 : \min _{x_*\in \{-1,1\}} \Vert X(t;x) - x_*\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0 \right\}$. Let $T_0 \ge 1$ to be fixed below and notice that for any $T_1>0$

$$\begin{aligned} \mathbb {P}\left( g(\sigma (x))^{\frac{1}{p_0}} \ge m T_0\right)&\le \mathbb {P}\left( g(\sigma (x))^{\frac{1}{p_0}} \ge m T_0, \, \sigma (x)\le m T_1\right) + \mathbb {P}(\sigma (x) \ge m T_1) \\&\le \mathbb {P}\left( \sum _{k=1}^{\lfloor m T_1 \rfloor } \sum _{l= 0,\frac{1}{2}} L_k(l,mT_1) \ge m (T_0 - C)\right) \\&\quad + \mathbb {P}(\sigma (x) \ge m T_1), \end{aligned}$$

for some $C>0$, where in the second inequality we use convexity of the mapping $g\mapsto g^{\frac{1}{p_0}}$ and the fact that $L_k(l,\sigma )$ is increasing in $\sigma$ by Definition 4.3. By Proposition 6.6 we can choose $T_1>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {P}(\sigma (x) \ge m T_1) \le \left( 1- \mathrm {e}^{-b/\varepsilon }\right) ^m. \end{aligned}$$

We also notice that

$$\begin{aligned}&\mathbb {P}\left( \sum _{k=1}^{\lfloor m T_1 \rfloor } \sum _{l= 0,\frac{1}{2}} L_k(l,mT_1) \ge m (T_0 - C)\right) \\&\quad \le \sum _{l=0,\frac{1}{2}} \mathbb {P}\left( \sum _{k=1}^{\lfloor m T_1 \rfloor }L_k(l,l+k) \ge m \left( \frac{T_0 - C}{2}\right) \right) \\&\quad \le \sum _{l=0,\frac{1}{2}}\exp \left\{ -c m \left( \frac{T_0 - C}{2\varepsilon }\right) \right\} \left( \mathbb {E}\mathrm {e}^{cL_1(l,1)/\varepsilon }\right) ^{mT_1}, \end{aligned}$$

where in the first inequality we use that $L_k(l,mT_1) \le L_k(l,l+k)$, for every $1\le k\le \lfloor m T_1 \rfloor$, and in the second we use an exponential Chebyshev inequality, independence and equality in law of the $L_k(l,l+k)$’s. For any $T>0$ we choose $c\equiv c(n)>0$ according to Proposition D.1, $T_0$ sufficiently large and $\varepsilon _0\in (0,1)$ sufficiently small such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sum _{l=0,\frac{1}{2}}\exp \left\{ -c m \left( \frac{T_0 - C}{2\varepsilon }\right) \right\} \left( \mathbb {E}\mathrm {e}^{cL_1(l,1)/\varepsilon }\right) ^{mT_1} \le \mathrm {e}^{-mT/\varepsilon }. \end{aligned}$$

Combining all the previous inequalities imply that

$$\begin{aligned}&\sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}\left( \left[ L(\nu _1(x),\rho _1(x);\sigma _1(x)) + (2-\kappa ) \sigma _1(x) + M_0\right] ^\frac{1}{p_0} \ge m T_0\right) \\&\quad \le \mathrm {e}^{-m T/\varepsilon } + \left( 1 - \mathrm {e}^{-b/\varepsilon }\right) ^m. \end{aligned}$$

This completes the proof if we relabel b since T is arbitrary. $\square$

6.3 Proof of Proposition 4.7

In this section we set

$$\begin{aligned}&f_i(x) := \frac{\kappa }{2} \tau _i(x) \\&g_i(x) := L(\nu _i(x),\rho _i(x);\sigma _i(x)) + (2-\kappa ) \sigma _i(x) + M_0. \end{aligned}$$

In this notation the random walk $S_N(x)$ in Definition 4.5 is given by $\sum _{i\le N} (f_i(x) - g_i(x))$.

To prove Proposition 4.7 we first consider a sequence of i.i.d. random variables $\{{\tilde{f}}_i\}_{i\ge 1}$ such that $\tilde{f}_1 \sim \exp (1)$. We furthermore assume that the family $\{\tilde{f}_i\}_{i\ge 1}$ is independent from both $\{f_i(x)\}_{i\ge 1}$ and $\{g_i(x)\}_{i\ge 1}$. For $\lambda >0$ which we fix later on, we set

$$\begin{aligned} {\tilde{S}}_N(x) := \lambda \sum _{i\le N} {\tilde{f}}_i - \sum _{i\le N} g_i(x). \end{aligned}$$

In the proof of Proposition 4.7 below we compare the random walk $S_N(x)$ with ${\tilde{S}}_N(x)$. The idea is that $\sum _{i\le N} f_i(x)$ behaves like $\lambda \sum _{i\le N} \tilde{f}_i$ for suitable $\lambda >0$.

In the next proposition we estimate the new random walk $\tilde{S}_N(x)$ using stochastic dominance. In particular we assume that the family of random variables $\{g_i(x)\}_{i\ge 1}$ is stochastically dominated by a family of i.i.d. random variables $\{\tilde{g}_i\}_{i\ge 1}$ which does not depend on x and obtain a lower bound on $\mathbb {P}(-{\tilde{S}}_N(x) \le u \text { for every } N\ge 1)$.

From now on we denote by $\mu _Z$ the law of a random variable Z.

Proposition 6.8

Assume that there exists a family of i.i.d. random variables $\{{\tilde{g}}_i\}_{i\ge 1}$, independent from both $\{g_i(x)\}_{i\ge 1}$ and $\{{\tilde{f}}_i\}_{i \ge 1}$, such that

$$\begin{aligned} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0}\mathbb {P}(g_i(x) \ge g) \le \mathbb {P}({\tilde{g}}_i \ge g), \end{aligned}$$

for every $g\ge 0$. Let ${\tilde{S}}_N = \lambda \sum _{i\le N} \tilde{f}_i - \sum _{i\le N} {\tilde{g}}_i$. Then

$$\begin{aligned} \inf _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0}\mathbb {P}(-{\tilde{S}}_N(x) \le u \text { for every } N\ge 1) \ge \mathbb {P}(-{\tilde{S}}_N \le u \text { for every } N \ge 1). \end{aligned}$$

Proof

Let

$$\begin{aligned}&G_N(x,u) = \mathbb {P}(-{\tilde{S}}_M(x) \le u \text { for every } N \ge M \ge 1) \\&G_N(u) = \mathbb {P}(-{\tilde{S}}_M \le u \text { for every } N \ge M \ge 1). \end{aligned}$$

We first prove that for every $N\ge 1$ and every x

$$\begin{aligned} G_N(x,u) \ge G_N(u). \end{aligned}$$

(6.8)

For $N = 1$ we have that

$$\begin{aligned} G_1(x,u)&= \mathbb {P}(-\lambda {\tilde{f}}_1 + g_1(x) \le u) = \int _0^\infty \mathbb {P}(g_1(x) \le u+ \lambda f) \, \mu _{{\tilde{f}}_1}(\,\mathrm {d}f) \\&\ge \int _0^\infty \mathbb {P}({\tilde{g}}_1 \le u+\lambda f) \, \mu _{{\tilde{f}}_1}(\,\mathrm {d}f) = \mathbb {P}(-\lambda {\tilde{f}}_1 + {\tilde{g}}_1 \le u) = G_1(u). \end{aligned}$$

Let us assume that (6.8) holds for N. Let $\partial B_0 = \{y: \Vert y- (\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} =\delta _0\}$. Conditioning on $\left( {\tilde{f}}_1, g_1(x), X(\nu _2(x);x)\right)$ and using independence of ${\tilde{f}}_1$ from the joint law of $\left( g_1(x), X(\nu _2(x);x)\right)$ we notice that

$$\begin{aligned}&G_{N+1}(x,u)\nonumber \\&\quad = \int _0^\infty \int _{[0,u+\lambda f]\times \partial B_0} G_N(y,u+\lambda f-g) \, \mu _{(g_1(x),X(\nu _2(x);x))}(\,\mathrm {d}g, \,\mathrm {d}y) \, \mu _{\tilde{f}_1}(\,\mathrm {d}f) \nonumber \\&\quad {\mathop {\ge }\limits ^{(6.8)}} \int _0^\infty \int _{[0,u+\lambda f]\times \partial B_0} G_N(u+\lambda f-g) \, \mu _{(g_1(x),X(\nu _2(x);x))}(\,\mathrm {d}g, \,\mathrm {d}y) \, \mu _{\tilde{f}_1}(\,\mathrm {d}f) \nonumber \\&\quad = \int _0^\infty \int _{[0,u+\lambda f]} G_N(u+\lambda f-g) \, \mu _{g_1(x)}(\,\mathrm {d}g) \, \mu _{{\tilde{f}}_1}(\,\mathrm {d}f). \end{aligned}$$

(6.9)

In the last equality above we use that $G_N(u+\lambda f-g)$ does not depend on y, hence we can drop the integral with respect to y. Let

$$\begin{aligned} H(g) = {\mathbf {1}}_{\{g\le u+ \lambda f\}} G_N(u+\lambda f - g). \end{aligned}$$

Then for fixed $u,f\ge 0, H$ is decreasing with respect to g. By Lemma E.1

$$\begin{aligned} \int _{[0,u+\lambda f]} G_N(u+\lambda f-g) \, \mu _{g_1(x)}(\,\mathrm {d}g)&= \int H(g) \mu _{g_1(x)}(\,\mathrm {d}g) \ge \int H(g) \mu _{{\tilde{g}}_1}(\,\mathrm {d}g) \\&= \int _{[0,u+\lambda f]} G_N(u+\lambda f-g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g). \end{aligned}$$

Integrating the last inequality with respect to f with $\mu _{{\tilde{f}}_1}$ and combining with (6.9) we obtain

$$\begin{aligned} G_{N+1}(x,u) \ge \int _0^\infty \int _{[0,u+\lambda f]} G_N(u+\lambda f-g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g) \, \mu _{{\tilde{f}}_1}(\,\mathrm {d}f) = G_{N+1}(u), \end{aligned}$$

which proves (6.8). If we now take $N\rightarrow \infty$ in (6.8) we get for arbitrary x

$$\begin{aligned} G(x,u) \ge G(u), \end{aligned}$$

which completes the proof. $\square$

In the next proposition we prove existence of a family of random variables $\{{\tilde{g}}_i\}_{i\ge 1}$ that satisfy the assumption of Proposition 6.8 and estimate their first moment.

Proposition 6.9

There exists a family of i.i.d. random variables $\{\tilde{g}_i\}_{i\ge 1}$, independent from both $\{g_i(x)\}_{i\ge 1}$ and $\{{\tilde{f}}_i\}_{i\ge 1}$, such that

$$\begin{aligned} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0}\mathbb {P}(g_i(x) \ge g) \le \mathbb {P}({\tilde{g}}_i \ge g), \end{aligned}$$

and furthermore for every $b>0$ there exist $\varepsilon _0\in (0,1)$ and $C>0$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \mathbb {E}{\tilde{g}}_1 \le C \mathrm {e}^{b/\varepsilon }. \end{aligned}$$

Proof

We first notice that by the Markov property

$$\begin{aligned} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0}\mathbb {P}(g_i(x) \ge g) \le \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0 }\mathbb {P}(g_1(x) \ge g). \end{aligned}$$

Let F(g) be the right continuous version of the increasing function $1 - \sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {P}(g_1(x) \ge g)$. We consider a family of i.i.d. random variables such $\{\tilde{g}_i\}_{i\ge 1}$ independent from both $\{g_i(x)\}_{i\ge 1}$ and $\{{\tilde{f}}_i\}_{i\ge 1}$ such that $\mathbb {P}({\tilde{g}}_i\le g) = F(g)$. To estimate $\mathbb {E}{\tilde{g}}_1$ let $c_\varepsilon >0$ to be fixed below. We notice that

$$\begin{aligned} \mathbb {E}{\tilde{g}}_1 \le \sup _{g\ge 0} g \mathrm {e}^{-c_\varepsilon g^\frac{1}{p_0}} \mathbb {E}\exp \left\{ c_\varepsilon {\tilde{g}}_1^\frac{1}{p_0}\right\} \le \left( \frac{p_0\mathrm {e}^{-1}}{c_\varepsilon }\right) ^{p_0} \mathbb {E}\exp \left\{ c_\varepsilon {\tilde{g}}_1^\frac{1}{p_0}\right\} . \end{aligned}$$

(6.10)

For $b>0$ we choose $T_0>0$ and $\varepsilon _0\in (0,1)$ as in Proposition 6.7. Then for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \mathbb {E}\exp \left\{ c_\varepsilon {\tilde{g}}_1^\frac{1}{p_0}\right\}&= 1 + \int _0^\infty c_\varepsilon \mathrm {e}^{c_\varepsilon g} \mathbb {P}\left( {\tilde{g}}_1^\frac{1}{p_0} \ge g\right) \,\mathrm {d}g \\&\le 1+ \sum _{m\ge 0} \mathbb {P}\left( {\tilde{g}}_1^\frac{1}{p_0} \ge mT_0\right) \int _{mT_0}^{(m+1)T_0} c_\varepsilon \mathrm {e}^{c_\varepsilon g} \,\mathrm {d}g \\&= 1+ \sum _{m\ge 0} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}\left( g_1(x)^\frac{1}{p_0} \ge mT_0\right) \int _{mT_0}^{(m+1)T_0} c_\varepsilon \mathrm {e}^{c_\varepsilon g} \,\mathrm {d}g \\&\le 1 + \mathrm {e}^{c_\varepsilon T_0} \sum _{m\ge 0} \mathrm {e}^{mc_\varepsilon T_0} \left( 1-\mathrm {e}^{-b/\varepsilon }\right) ^m, \end{aligned}$$

where in the last inequality we use Proposition 6.7 to estimate $\mathbb {P}\left( g_1(x)^\frac{1}{p_0} \ge mT_0\right)$. We now choose $c_\varepsilon >0$ such that $c_\varepsilon T_0 = \log \left( 1+\mathrm {e}^{-b/\varepsilon }\right)$. Then

$$\begin{aligned} \mathbb {E}\exp \left\{ c_\varepsilon {\tilde{g}}_1^\frac{1}{p_0}\right\}&\le 1 + \left( 1+\mathrm {e}^{-b/\varepsilon }\right) \sum _{m\ge 0} \left( 1+\mathrm {e}^{-b/\varepsilon }\right) ^m \left( 1-\mathrm {e}^{-b/\varepsilon }\right) ^m\\&\le 1+2 \sum _{m\ge 0} \left( 1-\mathrm {e}^{-2b/\varepsilon }\right) ^m \\&= 1+ 2 \mathrm {e}^{2b/\varepsilon }. \end{aligned}$$

Finally, by (6.10) we obtain that

$$\begin{aligned} \mathbb {E}{\tilde{g}}_1 \le \left( \frac{p_0\mathrm {e}^{-1}T_0}{\log \left( 1+\mathrm {e}^{-b/\varepsilon }\right) }\right) ^{p_0} \left( 1+ 2\mathrm {e}^{2b/\varepsilon }\right) , \end{aligned}$$

which completes the proof if we relabel b. $\square$

Remark 6.10

In the proof of Proposition 6.9 we use stretched exponential moments of ${\tilde{g}}_1$, although we only need 1st moments (see Lemma 6.12 below). This simplifies our calculations.

From now on we let ${\tilde{S}}_N = \lambda \sum _{i\le N} {\tilde{f}}_i - \sum _{i\le N} {\tilde{g}}_i$ for $\{{\tilde{g}}_i\}_{i\ge 1}$ as in Proposition 6.9.

In the next proposition we explicitly compute the probability $\mathbb {P}(-{\tilde{S}}_N \le 0 \text { for every } N\ge 1)$. The proof is essentially the same as the classical Cramér–Lundberg estimate (see [8, Chapter 1.2]). We present it here for the reader’s convenience.

Proposition 6.11

For the random walk ${\tilde{S}}_N$ the following estimate holds,

$$\begin{aligned} \mathbb {P}(-{\tilde{S}}_N \le 0 \text { for every } N \ge 1) = 1 - \frac{1}{\lambda } \mathbb {E}{\tilde{g}}_1. \end{aligned}$$

Proof

Let $G(u) = \mathbb {P}(-{\tilde{S}}_N \le u \text { for every } N \ge 1)$. Conditioning on $({\tilde{f}}_1,{\tilde{g}}_1)$ and using independence we notice that

$$\begin{aligned} G(u)&= \mathbb {P}\left( -\lambda \sum _{i=2}^N {\tilde{f}}_i + \sum _{i=2}^N {\tilde{g}}_i \le u + \lambda {\tilde{f}}_1 - {\tilde{g}}_1 \text { for every } N\ge 2, \, -\lambda {\tilde{f}}_1 + {\tilde{g}}_1 \le u\right) \nonumber \\&= \int _0^\infty \int _0^{u+\lambda f} G(u+\lambda f-g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g) \, \mu _{{\tilde{f}}_1}(\,\mathrm {d}f) \nonumber \\&= \frac{1}{\lambda } \mathrm {e}^{u/\lambda } \int _u^\infty \mathrm {e}^{-{\bar{f}}/\lambda } \int _0^{{\bar{f}}} G({\bar{f}}-g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g) \,\mathrm {d}{\bar{f}}, \end{aligned}$$

(6.11)

where in the last equality we use that ${\tilde{f}}_1\sim \exp (1)$ and we also make the change of variables ${\bar{f}} = u+\lambda f$. This implies that G(u) is differentiable with respect to u and in particular

$$\begin{aligned} \partial _{{\bar{u}}} G({\bar{u}}) = \frac{1}{\lambda } G({\bar{u}}) - \frac{1}{\lambda } \int _0^{{\bar{u}}} G({\bar{u}}-g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g). \end{aligned}$$

Integrating the last equation form 0 to u we obtain that

$$\begin{aligned} G(u) = G(x,0) + \frac{1}{\lambda } \int _0^u G(u -{\bar{u}}) \,\mathrm {d}{\bar{u}} - \frac{1}{\lambda } \int _0^u \int _0^{{\bar{u}}} G({\bar{u}}-g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g) \,\mathrm {d}{\bar{u}}. \end{aligned}$$

(6.12)

Let $F(g) := \mu _{{\tilde{g}}_1}([0,g])$. A simple integration by parts implies

$$\begin{aligned}&\int _0^u \int _0^{{\bar{u}}} G({\bar{u}}-g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g) \,\mathrm {d}{\bar{u}} \nonumber \\&\quad = \int _0^u \left( \left[ G({\bar{u}} - g)\right] _{g=0}^{{\bar{u}}} + \int _0^{{\bar{u}}} \partial _g G({\bar{u}} - g) F(g) \,\mathrm {d}g\right) \,\mathrm {d}\bar{u} \nonumber \\&\quad = \int _0^u G(0) F({\bar{u}}) \,\mathrm {d}{\bar{u}} + \int _0^u \int _g^u \partial _g G({\bar{u}}-g) \,\mathrm {d}{\bar{u}} F(g) \,\mathrm {d}g \nonumber \\&\quad = \int _0^u G(0) F({\bar{u}}) \,\mathrm {d}{\bar{u}} - \int _0^u \left[ -G({\bar{u}} -g)\right] _g^u F(g) \,\mathrm {d}g \nonumber \\&\quad = \int _0^u G(u-g) F(g) \,\mathrm {d}g. \end{aligned}$$

(6.13)

Combining (6.12) and (6.13) we get

$$\begin{aligned} G(u) = G(0) + \frac{1}{\lambda } \int _0^u G(u - {\bar{u}}) \,\mathrm {d}{\bar{u}} - \frac{1}{\lambda } \int _0^u G(u - {\bar{u}} ) F({\bar{u}}) \,\mathrm {d}{\bar{u}}. \end{aligned}$$

By taking $u\rightarrow \infty$ in the last equation and using the dominated convergence theorem and the law of large numbers we finally obtain

$$\begin{aligned} 1 = G(0) + \frac{1}{\lambda } \mathbb {E}{\tilde{g}}_1, \end{aligned}$$

which completes the proof. $\square$

Combining Propositions 6.8, 6.9 and 6.11 we obtain the following lemma.

Lemma 6.12

For any $b>0$ there exist $\varepsilon _0\in (0,1)$ and $C>0$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \inf _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}(-{\tilde{S}}_N(x) \le 0 \text { for every } N\ge 1) \ge 1 - C \frac{\mathrm {e}^{b/\varepsilon }}{\lambda }. \end{aligned}$$

Proof

By Propositions 6.8, 6.9 and 6.11 and

$$\begin{aligned}&\inf _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0} \mathbb {P}(-{\tilde{S}}_N(x) \le 0 \text { for every } N\ge 1) \\&\quad \ge \mathbb {P}(-{\tilde{S}}_N \le 0 \text { for every } N\ge 1) = 1 - \frac{1}{\lambda } \mathbb {E}{\tilde{g}}_1. \end{aligned}$$

Moreover, by Proposition 6.9 for every $b>0$ there exist $\varepsilon _0\in (0,1)$ and $C>0$ such that for every $\varepsilon \le \varepsilon _0, \mathbb {E}{\tilde{g}}_1\le C \mathrm {e}^{b/\varepsilon }$ which completes the proof. $\square$

We are now ready to prove Proposition 4.7 which is the main goal of this section.

Proof of Proposition 4.7

We estimate $\mathbb {P}(S_N(x) \le 0 \text { for some } N\ge 1)$ in the following way,

$$\begin{aligned}&\mathbb {P}(-S_N(x) \ge 0 \text { for some } N\ge 1)\nonumber \\&\quad \le \mathbb {P}\left( -\sum _{i\le N} f_i(x) + \lambda \sum _{i\le N} {\tilde{f}}_i \ge 0 \text { for some } N \ge 1\right) \nonumber \\&\qquad + \mathbb {P}(-{\tilde{S}}_N(x) \ge 0 \text { for some } N \ge 1). \end{aligned}$$

(6.14)

The second term on the right hand side can be estimated by Lemma 6.12 which provides a bound of the form

$$\begin{aligned} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}(-{\tilde{S}}_N(x) \ge 0 \text { for some } N \ge 1) \le C \frac{\mathrm {e}^{b/\varepsilon }}{\lambda }. \end{aligned}$$

(6.15)

For the first term we notice that

$$\begin{aligned}&\mathbb {P}\left( -\sum _{i\le N} f_i(x) + \lambda \sum _{i\le N} \tilde{f}_i \ge 0 \text { for some } N \ge 1\right) \\&\quad \le \sum _{N\ge 1} \mathbb {P}\left( -\sum _{i\le N} f_i(x) + \lambda \sum _{i\le N} {\tilde{f}}_i \ge 0\right) \\&\quad \le \sum _{N\ge 1} \mathbb {P}\left( \exp \left\{ -\frac{1}{2\lambda }\sum _{i\le N} f_i(x) + \frac{1}{2} \sum _{i\le N} {\tilde{f}}_i\right\} \ge 1\right) . \end{aligned}$$

By Markov’s inequality, independence of $\{f_i(x)\}_{i\ge 1}$ and $\{{\tilde{f}}_i\}_{i\ge 1}$ and equality in law of the ${\tilde{f}}_i$’s the last inequality implies that

$$\begin{aligned}&\mathbb {P}\left( -\sum _{i\le N} f_i(x) + \lambda \sum _{i\le N} {\tilde{f}}_i \ge 0 \text { for some } N \ge 1\right) \nonumber \\&\quad \le \sum _{N\ge 1} \underbrace{\mathbb {E}\exp \left\{ -\frac{1}{2\lambda }\sum _{i\le N}f_i(x)\right\} }_{=:I_N(x)} \underbrace{\left( \mathbb {E}\exp \left\{ \frac{{\tilde{f}}_1}{2}\right\} \right) ^N}_{\le 2^N \text { since } {\tilde{f}}_1\sim \exp (1)}. \end{aligned}$$

(6.16)

Let $\varepsilon _0\in (0,1)$ as in Proposition 6.3. For the term $I_N(x)$ we notice that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&I_N(x) \le \left( \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0}\mathbb {E}\exp \left\{ -\frac{1}{2\lambda }f_1(x)\right\} \right) ^N \\&\quad \le \left( \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0}\left[ \mathbb {E}\exp \left\{ -\frac{1}{2\lambda }f_1(x)\right\} {\mathbf {1}}_{\{f_1(x) \ge \mathrm {e}^{2a_0/\varepsilon }\}} \right. \right. \\&\left. \left. \qquad + \mathbb {P}\left( f_1(x) \le \mathrm {e}^{2a_0/\varepsilon }\right) \right] \right) ^N \\&\quad \le \left( \mathrm {e}^{-\mathrm {e}^{2a_0/\varepsilon }/2\lambda } + \mathrm {e}^{-3a_0/\varepsilon }\right) ^N, \end{aligned}$$

where in the first inequality we use the Markov property and in the last we use Proposition 6.3. If we choose $\frac{1}{2\lambda }=\mathrm {e}^{-(2a_0-b)/\varepsilon }$ and choose $\varepsilon _0\in (0,1)$ even smaller the last inequality implies that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} I_N(x) \le \left( \mathrm {e}^{-\mathrm {e}^{b/\varepsilon }} + \mathrm {e}^{-3a_0/\varepsilon }\right) ^N \le \mathrm {e}^{-5a_0N/2\varepsilon }. \end{aligned}$$

Combining with (6.16) we find $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&\sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}\left( -\sum _{i\le N} f_i(x) + \lambda \sum _{i\le N} {\tilde{f}}_i \ge 0 \text { for some } N \ge 1\right) \nonumber \\&\quad \le \sum _{N\ge 1} \mathrm {e}^{-5a_0N/2\varepsilon } 2^N \le \sum _{N\ge 1} \mathrm {e}^{-2a_0N/\varepsilon } = \frac{\mathrm {e}^{-2a_0/\varepsilon }}{1-\mathrm {e}^{-2a_0/\varepsilon }}. \end{aligned}$$

(6.17)

Finally (6.14), (6.15) and (6.17) imply that

$$\begin{aligned} \sup _{\Vert x-(\pm 1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0}\mathbb {P}(-S_N(x) \ge 0 \text { for some } N\ge 1) \le C\frac{\mathrm {e}^{b/\varepsilon }}{\mathrm {e}^{(2a_0-b)/\varepsilon }} + \frac{\mathrm {e}^{-2a_0/\varepsilon }}{1- \mathrm {e}^{-2a_0/\varepsilon }}, \end{aligned}$$

which completes the proof since b is arbitrary. $\square$

7 Proof of the Eyring–Kramers law

In this section we prove the Eyring–Kramers law Theorem 3.5. We first need to introduce some additional tools.

We consider the spatial Galerkin approximation $X_N(\cdot ;x)$ of $X(\cdot ;x)$ given by

$$\begin{aligned}&(\partial _t - \Delta ) X_N = - \Pi _N\left( X_N^3 - X_N - 3 \varepsilon \mathfrak {R}_N X_N \right) + \sqrt{2\varepsilon } \xi _N \nonumber \\&X_N\big |_{t=0} = x_N \end{aligned}$$

(7.1)

where $\Pi _N$ is the projection on $\{f \in L^2: f(z)=\sum _{|k|\le N} {\hat{f}}(k) L^{-2}\mathrm {e}^{2\pi \mathrm {i}k\cdot z/L}\}, \xi _N = \Pi _N \xi , x_N = \Pi _N x$ and $\mathfrak {R}_N$ is as in (2.3). Here for $k\in \mathbb {Z}^2$ we set $|k|=|k_1|\vee |k_2|$. In this notation we have that $\Pi _N f = f*D_N$, where $D_N$ is the 2-dimensional square Dirichlet kernel given by $D_N(z) = \sum _{|k|\le N} L^{-2}\mathrm {e}^{2\pi \mathrm {i}k\cdot z/L}$.

To treat (7.1) we write for

Then $v_N(\cdot ;x)$ solves

(7.2)

where and .

As in (3.1) and (3.2), for $\delta \in (0,1/2)$ and $\alpha >0$ we define the symmetric subsets A and B of $\mathcal {C}^{-\alpha }$ by

$$\begin{aligned}&A := \left\{ f\in \mathcal {C}^{-\alpha } : {\bar{f}} \in [-1-\delta ,-1+\delta ], \, f - {\bar{f}} \in D_{\perp } \right\} \end{aligned}$$

(7.3)

$$\begin{aligned}&B := \left\{ f\in \mathcal {C}^{-\alpha } : {\bar{f}} \in [1-\delta ,1+\delta ], \, f - {\bar{f}} \in D_{\perp }\right\} \end{aligned}$$

(7.4)

where $D_\perp$ is the closed ball of radius $\delta$ in $\mathcal {C}^{-\alpha }$ and ${\bar{f}} = L^{-2} \langle f, 1 \rangle$. To simplify the notation in this section, we have dropped the dependence of A and B on the parameters $\alpha$ and $\delta$. We will only write $A(\alpha ;\delta )$ and $B(\alpha ;\delta )$ if we need to specify the values of these parameters. For $x\in A$ we define

$$\begin{aligned} \tau _B(X_N(\cdot ;x)) := \inf \left\{ t>0: X_N(t;x)\in B\right\} \end{aligned}$$

and

$$\begin{aligned} \tau _B(X(\cdot ;x)) := \inf \left\{ t>0: X(t;x)\in B\right\} . \end{aligned}$$

Last, recall that for $k\in \mathbb {Z}^2$ (see (3.3)),

$$\begin{aligned} \lambda _k = \left( \frac{2\pi |k|}{L}\right) ^2 -1 \text { and } \quad \nu _k = \left( \frac{2\pi |k|}{L}\right) ^2 +2 = \lambda _k + 3. \end{aligned}$$

The next theorem is essentially [2, Theorem 2.3].

Theorem 7.1

([2, Theorem 2.3]) Let $0< L < 2\pi$. For every $\alpha >0, \delta \in (0,1/2)$ and $\varepsilon \in (0,1)$ there exists a sequence $\{\mu _{\varepsilon ,N}\}_{N\ge 1}$ of probability measures concentrated on $\partial A$ such that

$$\begin{aligned}&\limsup _{N\rightarrow \infty }\int \mathbb {E}\tau _B(X_N(\cdot ;x)) \, \mu _{\varepsilon ,N}(\,\mathrm {d}x) \nonumber \\&\quad \le \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} } \mathrm {e}^{\left( V(0) - V(-1)\right) /\varepsilon }\left( 1+c_+\sqrt{\varepsilon }\right) \nonumber \\&\liminf _{N\rightarrow \infty }\int \mathbb {E}\tau _B(X_N(\cdot ;x)) \, \mu _{\varepsilon ,N}(\,\mathrm {d}x) \nonumber \\&\quad \ge \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} } \mathrm {e}^{\left( V(0) - V(-1)\right) /\varepsilon }\left( 1-c_-\varepsilon \right) \end{aligned}$$

(7.5)

where the constants $c_+$ and $c_-$ are uniform in $\varepsilon$.

Proof

The proof of (7.5) is given in [2, Sections 4 and 5], but the following should be modified.

• In [2], the sets A and B are defined as in (7.3) and (7.4) with $D_\perp$ replaced by a ball in $H^s$ for $s<0$. The explicit form of $D_\perp$ is only used in [2, Lemma 5.9]. There the authors consider the 0-mean Gaussian measure $\gamma ^\perp _0$ with quadratic form $\frac{1}{2\varepsilon } \left( \Vert \nabla f\Vert _{L^2}^2 - \Vert f-\bar{f}\Vert _{L^2}^2\right)$, and prove that $D_\perp$ has probability bounded from below by $1- c\varepsilon ^2$. Here we assume that $D_\perp$ is a ball in $\mathcal {C}^{-\alpha }$. To obtain the same estimate for this set, we first notice that the random field f associated with the measure $\gamma ^\perp _0$ satisfies

  $$\begin{aligned} \mathbb {E}\langle f, L^{-2}\mathrm {e}^{2\mathrm {i}\pi k \cdot /L} \rangle \lesssim \frac{\varepsilon \log \varepsilon ^{-1} \log \lambda _k}{1+\lambda _k}, \end{aligned}$$

  for every $k\in \mathbb {Z}^2$, where the explicit constant depends on L. This decay of the Fourier modes of f and [21, Proposition 3.6] imply that the measure $\gamma ^\perp _0$ is concentrated in $\mathcal {C}^{-\alpha }$, for every $\alpha >0$, which in turn implies [2, Lemma 5.9] for the set $D_\perp$ considered here.

• In [2], the authors consider (7.1) with $\mathfrak {R}_N$ replaced by

  $$\begin{aligned} C_N = \frac{1}{L^2} \sum _{|k|\le N} \frac{1}{|\lambda _k|} \end{aligned}$$

  and obtain (7.5) with the pre-factor given by

  $$\begin{aligned} \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k}\right\} } = \lim _{N\rightarrow \infty } \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{|k|\le N}\frac{|\lambda _k|}{\nu _k}} \exp \left\{ \frac{3 L^2 C_N}{2}\right\} . \end{aligned}$$

  In our case one can check by (2.3) that $\mathfrak {R}_N$ is given by

  $$\begin{aligned} \mathfrak {R}_N = \frac{1}{L^2} \sum _{|k|\le N} \frac{1}{|\lambda _k+2|}. \end{aligned}$$

  According to [2, Remark 2.5] this choice of renormalisation constant modifies [2, Theorem 2.3] by multiplying the pre-factor there with

  $$\begin{aligned} \exp \left\{ -3 L^2 \lim _{N\rightarrow \infty } (\mathfrak {R}_N - C_N)/2\lambda _0\right\} . \end{aligned}$$

$\square$

Remark 7.2

The finite dimensional measure $\mu _{\varepsilon ,N}$ in (7.5) is given by

$$\begin{aligned} \mu _{\varepsilon ,N}(\,\mathrm {d}f) = \frac{1}{\mathrm {cap}_A(B)} \mathrm {e}^{-V(\Pi _Nf)/\varepsilon } \rho _{A,B}( \,\mathrm {d}f), \end{aligned}$$

where $\rho _{A,B}$ is a probability measure concentrated on $\partial A$, called the equilibrium measure, and $\mathrm {cap}_A(B)$ is a normalisation constant. Under this measure and the assumption that the sets A and B are symmetric, the integrals appearing in (7.5) can be rewritten using potential theory as

$$\begin{aligned} \int \mathbb {E}\tau _B(X_N(\cdot ;x)) \, \mu _{\varepsilon ,N}(\,\mathrm {d}x) = \frac{1}{2\mathrm {cap}_A(B)} \int _{\mathbb {R}^{(2N+1)^2}} \mathrm {e}^{-V(\Pi _Nf)/\varepsilon } \,\mathrm {d}f. \end{aligned}$$

This formula is derived in [2, Section 3] and it is then analysed to obtain (7.5).

Theorem 3.5 generalises (7.5) for the limiting process $X(\cdot ;x)$ for fixed initial condition x in a suitable neighbourhood of $-1$. To prove this theorem, we first fix $\alpha \in (0,\alpha _0)$ and pass to the limit as $N\rightarrow \infty$ in (7.5) to prove a version of (3.4) where the initial condition x is averaged with respect to a measure $\mu _\varepsilon$ concentrated on a closed ball with respect to the weaker topology $\mathcal {C}^{-\alpha _0}$ (see Proposition 7.7). This measure is the weak limit, up to a subsequence, of the measures $\mu _{\varepsilon ,N}$ in Theorem 7.1. We then use the exponential loss of memory, Theorem 3.1, to pass from averages of initial conditions with respect to the limiting measure $\mu _\varepsilon$ to fixed initial conditions.

The rest of this section is structured as follows. In Sect. 7.1 we prove convergence of the Galerkin approximations $X_N(\cdot ;x_N)$ and obtain estimates uniform in the initial condition x and the regularisation parameter N. In Sect. 7.2 we prove uniform integrability of the stopping times $\tau _B(X(\cdot ;x))$ and pass to the limit as $N\rightarrow \infty$ in (7.5). Finally in Sect. 7.3 we prove Theorem 3.5.

7.1 Convergence of the Galerkin scheme and a priori estimates

In the next proposition we prove convergence of $X_N(\cdot ;x)$ to $X(\cdot ;x_N)$ in $C([0,T];\mathcal {C}^{-\alpha })$ using convergence of the stochastic objects which is proven in [24, Proposition 2.3]. This is a technical result and the proof is given in the Appendix.

Proposition 7.3

Let $\aleph \subset \mathcal {C}^{-\alpha _0}$ be bounded and assume that for every $x\in \aleph$, there exists a sequence $\{x_N\}_{N\ge 1}$ such that $x_N\rightarrow x$ uniformly in x. Then for every $\alpha \in (0,\alpha _0)$ and $0<s<T$

$$\begin{aligned} \lim _{N\rightarrow \infty } \sup _{x\in \aleph } \sup _{t\in [s,T]} \Vert X_N(t;x_N) - X(t;x)\Vert _{\mathcal {C}^{-\alpha }} = 0 \end{aligned}$$

in probability.

Proof

See Appendix F. $\square$

The next proposition provides a bound for $X_N(\cdot ;x)$ uniformly in the initial condition x and the regularisation parameter N in the $\mathcal {B}^{-\alpha }_{2,2}$ norm, for $0<\alpha <\alpha _0$. This result has been already established in [24, Corollary 3.10] for the limiting process $X(\cdot ;x)$ in the $\mathcal {C}^{-\alpha }$ norm. There (2.2) is tested with $v(\cdot ;x)^{p-1}$, for $p\ge 2$ even, to bound $\Vert v(\cdot ;x)\Vert _{L^p}$ by using the “good” sign of the non-linear term $-v^3$. In the case of (7.2) this argument allows us to bound $\Vert v_N(\cdot ;x)\Vert _{L^p}$ for $p=2$ only, because of the projection $\Pi _N$ in front of the non-linearity.

Proposition 7.4

For every $\alpha \in (0,\alpha _0]$ and $p\ge 1$ we have that

$$\begin{aligned} \sup _{N\ge 1}\sup _{x\in \mathcal {C}^{-\alpha _0}}\sup _{t\le 1} t^{\frac{p}{2}} \mathbb {E}\Vert X_N(s;x)\Vert _{\mathcal {B}^{-\alpha }_{2,2}}^p <\infty . \end{aligned}$$

(7.6)

Proof

Proceeding exactly as in the proof of [24, Proposition 3.7] we first show that there exist $\alpha \in (0,1)$ and $p_n\ge 1$ such that for every $t\in (0,1)$

(7.7)

for every $\alpha '\in (0,1)$, uniformly in $x\in \mathcal {C}^{-\alpha _0}$. We then proceed as in the proof of [24, Corollary 3.10] and use (7.7) to prove (7.6). The only difference is that here we use the norm $\Vert \cdot \Vert _{\mathcal {B}^{-\alpha }_{2,2}}$ and the embedding $L^2\hookrightarrow \mathcal {B}^{-\alpha }_{2,2}$ on the level of $v_N(\cdot ;x)$ together with the fact that

for every $\alpha ,\alpha '>0$ and $p\ge 1$, which is immediate from [24, Proposition 2.2, Proposition 2.3]. $\square$

7.2 Passing to the limit

In this section we pass to the limit as $N\rightarrow \infty$ in (7.5) using uniform integrability of the stopping time $\tau _B(X_N(\cdot ;x))$. To obtain uniform integrability we prove exponential moment bounds for $\tau _B(X_N(\cdot ;x))$ uniformly in the initial condition $x \in \mathcal {C}^{-\alpha _0}$ and the regularisation parameter N. We first bound $\mathbb {P}\left( \tau _B(X_N(\cdot ;x)) \ge 1\right)$ using a support theorem and a strong a priori bound for $X_N(\cdot ;x)$ in $\mathcal {C}^{-\alpha }$. A support theorem for the limiting process $X(\cdot ;x)$ has been already established in [24, Corollary 6.4]. To use it for $X_N(\cdot ;x)$ we combine it with the convergence result in Proposition 7.3. To obtain a strong a priori bound for $X_N(\cdot ;x)$ in $\mathcal {C}^{-\alpha }$ we first use Proposition 7.4 which implies the bound in $\mathcal {B}^{-\alpha }_{2,2}$ and then use Proposition G.2 to pass from the $\mathcal {B}^{-\alpha }_{2,2}$ norm to the $\mathcal {C}^{-\alpha }$ norm.

Proposition 7.5

For every $\alpha \in (0,\alpha _0), \delta \in (0,1/2)$ and $\varepsilon \in (0,1)$ there exist $c_0 \in (0,1)$ and $N_0\ge 1$ such that for every $N\ge N_0$

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}}\mathbb {P}\left( \tau _B(X_N(\cdot ;x)) \ge 1\right) \le c_0. \end{aligned}$$

Proof

Let $\alpha \in (0,\alpha _0)$ and let $\aleph$ be a compact subset of $\mathcal {C}^{-\alpha _0}$ which we fix below. Using the Markov property

$$\begin{aligned}&\mathbb {P}(\tau _B(X_N(\cdot ;x))\ge 1) \le \sup _{y\in \aleph } \mathbb {P}(\tau _B(X_N(\cdot ;y))\ge 1/2) \, \mathbb {P}(X_N(1/2;x)\in \aleph ) \\&\quad + \mathbb {P}(X_N(1/2;x)\notin \aleph ). \end{aligned}$$

The proof is complete if for every $N\ge N_0$

$$\begin{aligned} \sup _{y\in \aleph } \mathbb {P}(\tau _B(X_N(\cdot ;y))\ge 1/2)< 1, \, \sup _{x\in \mathcal {C}^{-\alpha _0}}\mathbb {P}(X_N(1/2;x)\notin \aleph ) <1. \end{aligned}$$

(7.8)

We notice that there exists $\delta '>0$ such that for any $y\in \aleph$

$$\begin{aligned} \mathbb {P}(\tau _B(X_N(\cdot ;y))\le 1/2)&\ge \mathbb {P}(X_N(1/2;y)\in B) \ge \mathbb {P}\left( X(1/2;y)\in B_{\mathcal {C}^{-\alpha }}(1;\delta ')\right) \nonumber \\&\quad - \mathbb {P}\left( \Vert X_N(1/2;y) - X(1/2;y)\Vert _{\mathcal {C}^{-\alpha }} \ge \delta '\right) . \end{aligned}$$

(7.9)

Here we use that if $\Vert X(1/2;y)-1\Vert _{\mathcal {C}^{-\alpha }},\Vert X_N(1/2;y) - X(1/2;y)\Vert _{\mathcal {C}^{-\alpha }} \le \delta '$, then $X_N(1/2;y) \in B$ for $\delta '$ sufficiently small. By the support theorem [24, Corollary 6.4] there exists $c_1 \equiv c_1(\delta ,\varepsilon )>0$ such that

$$\begin{aligned} \inf _{y\in \aleph }\mathbb {P}\left( X(1/2;y) \in B_{\mathcal {C}^{-\alpha }}(1; \delta ')\right) \ge c_1. \end{aligned}$$

(7.10)

On the other hand Proposition 7.3 implies convergence in probability of $X_N(1/2;y)$ to X(1 / 2; y) in $\mathcal {C}^{-\alpha }$ uniformly in $y\in \aleph$. Hence there exists $N_0\ge 1$ such that for every $N\ge N_0$

$$\begin{aligned} \sup _{y\in \aleph }\mathbb {P}\left( \Vert X_N(1/2;y) - X(1/2;y)\Vert _{\mathcal {C}^{-\alpha }} \ge \delta /2\right) \le c_1/2. \end{aligned}$$

(7.11)

Plugging (7.10) and (7.11) in (7.9) implies the first bound in (7.8).

We now prove the second bound in (7.8). By the Markov inequality for every $R>0$

$$\begin{aligned} \mathbb {P}\left( \Vert X_N(1/4;x)\Vert _{\mathcal {B}^{-\alpha }_{2,2}} \ge R\right) \le \frac{1}{R} \mathbb {E}\Vert X_N(1/4;x)\Vert _{\mathcal {B}^{-\alpha }_{2,2}}. \end{aligned}$$

By (7.6) the expectation on the right hand side of the last inequality is uniformly bounded over $x\in \mathcal {C}^{-\alpha _0}$ and $N\ge 1$. Thus choosing $R>0$ large enough

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {P}\left( \Vert X_N(1/4;x)\Vert _{\mathcal {B}^{-\alpha }_{2,2}} \ge R\right) \le \frac{1}{2}. \end{aligned}$$

(7.12)

By Proposition G.2 for every $K,R>0$ there exist $C\equiv C(K,R)$ such that

Choosing K sufficiently large, combining the last inequality with [24, Propositions 2.2 and 2.3] and using the Markov inequality imply that

$$\begin{aligned} \sup _{\Vert y\Vert _{\mathcal {B}^{-\alpha }_{2,2}}\le R} \mathbb {P}\left( \Vert X_N(1/4;y)\Vert _{\mathcal {C}^{-\alpha }} \ge C\right) \le \frac{1}{2}. \end{aligned}$$

(7.13)

Using the Markov property and (7.12) and (7.13) we get for arbitrary $x\in \mathcal {C}^{-\alpha _0}$

$$\begin{aligned}&\mathbb {P}\left( \Vert X_N(1/2;x)\Vert _{\mathcal {C}^{-\alpha }} \ge C\right) \\&\quad \le \mathbb {P}\left( \Vert X_N(1/4;x)\Vert _{\mathcal {B}^{-\alpha }_{2,2}} \le R\right) \sup _{y\in \mathcal {B}^{-\alpha }_{2,2}} \mathbb {P}(\Vert X_N(1/4;y)\Vert _{\mathcal {C}^{-\alpha }} \ge C) \\&\qquad + \mathbb {P}\left( \Vert X_N(1/4;x)\Vert _{\mathcal {B}^{-\alpha }_{2,2}} \ge R\right) \le \frac{3}{4}. \end{aligned}$$

We finally notice that for every $\alpha <\alpha _0$ the set $\aleph = \{f\in \mathcal {C}^{-\alpha _0}: \Vert f\Vert _{\mathcal {C}^{-\alpha }}\le C\}$ is compact in $\mathcal {C}^{-\alpha _0}$ which implies the second bound in (7.8). $\square$

In the next corollary we use Proposition 7.5 to prove exponential moments for the stopping time $\tau _B(X_N(\cdot ;x))$.

Corollary 7.6

For every $\delta >0$ and $\varepsilon \in (0,1)$ there exist $\eta _0>0$ and $N_0\ge 1$ such that

$$\begin{aligned} \sup _{N\ge N_0}\sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {E}\exp \{\eta _0 \tau _B(X_N(\cdot ;x))\} <\infty . \end{aligned}$$

Proof

By the Markov property we have that

$$\begin{aligned} \mathbb {P}(\tau _B(X_N(\cdot ;x)) \ge k+1) \le \sup _{y\in \mathcal {C}^{-\alpha _0}} \mathbb {P}(\tau _B(X_N(\cdot ;y))\ge 1) \, \mathbb {P}(\tau _B(X_N(\cdot ;x))\ge k). \end{aligned}$$

Iterating this inequality and using Proposition 7.5 we obtain that

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {P}(\tau _B(X_N(\cdot ;x)) \ge k+1) \le c_0^{k+1}. \end{aligned}$$

Then

$$\begin{aligned} \mathbb {E}\exp \{\eta _0 \tau _B(X_N(\cdot ;x))\}&= 1 + \int _0^\infty \eta _0 \mathrm {e}^{\eta _0 t} \mathbb {P}(\tau _B(X_N(\cdot ;x)) \ge t) \,\mathrm {d}t \\&\le 1 + \sum _{k=0}^\infty \mathbb {P}(\tau _B(X_N(\cdot ;x)) \ge k) \int _k^{k+1} \eta _0 \mathrm {e}^{\eta _0 t} \,\mathrm {d}t \\&\le 1 + \mathrm {e}^{\eta _0} \sum _{k=0}^\infty \mathrm {e}^{\eta _0 k} c_0^k \end{aligned}$$

and the proof is complete if we choose $\eta _0<\log c_0^{-1}$. $\square$

In the next proposition we pass to the limit as $N\rightarrow \infty$ in (7.5). Here we use Corollary 7.6, which implies uniform integrability of $\tau _B(X_N(\cdot ;x))$, and the weak convergence of the measures $\mu _{\varepsilon ,N}$.

Proposition 7.7

For every $\alpha \in (0,\alpha _0), \delta \in (0,1/2)$ except possibly a countable subset, and $\varepsilon \in (0,1)$ there exists a probability measure $\mu _\varepsilon \in \mathcal {M}_1\left( A(\alpha _0;\delta )\right)$ such that

$$\begin{aligned}&\int \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \, \mu _\varepsilon (\,\mathrm {d}x) \nonumber \\&\quad \le \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} } \mathrm {e}^{\left( V(0) - V(-1)\right) /\varepsilon }\left( 1+c_+\sqrt{\varepsilon }\right) \nonumber \\&\int \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \, \mu _\varepsilon (\,\mathrm {d}x) \nonumber \\&\quad \ge \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} } \mathrm {e}^{\left( V(0) - V(-1)\right) /\varepsilon }\left( 1-c_-\varepsilon \right) \end{aligned}$$

(7.14)

where the constants $c_+$ and $c_-$ are uniform in $\varepsilon$.

Proof

We only prove the upper bound in 7.14. The lower bound follows similarly.

Let $\alpha \in (0,\alpha _0)$ and $\delta \in (0,1/2)$. Using the compact embedding $\mathcal {C}^{-\alpha } \hookrightarrow \mathcal {C}^{-\alpha _0}$ (see Proposition A.8), for any $\alpha <\alpha _0$, we have that $A(\alpha ;\delta ) \subset A(\alpha _0;\delta )$. Let $\{\mu _{\varepsilon ,N}\}_{N\ge 1}$ be the family of probability measures in (7.5). Using again the compact embedding $\mathcal {C}^{-\alpha } \hookrightarrow \mathcal {C}^{-\alpha _0}$, for any $\alpha <\alpha _0$, this family is trivially tight since it is concentrated on $\partial A(\alpha ;\delta )$. Hence there exists $\mu _\varepsilon \in \mathcal {M}_1\left( A(\alpha _0;\delta )\right)$ such that $\mu _{\varepsilon ,N} {\mathop {\rightarrow }\limits ^{\text {weak}}} \mu _\varepsilon$ up to a subsequence.

By Skorokhod’s represantation theorem (see [7, Theorem 2.4]) there exist a probability space $(\Omega _\mu , \mathcal {F}_\mu , \mathbb {P}_\mu )$ and random variables $\{x_N\}_{N\ge 1}$ and x taking values in $A(\alpha _0;\delta )$ such that $x_N{\mathop {=}\limits ^{\text {law}}} \mu _N, x{\mathop {=}\limits ^{\text {law}}} \mu _\varepsilon$ and $x_N\rightarrow x$$\mathbb {P}_{\mu _\varepsilon }$-almost surely in $\mathcal {C}^{-\alpha _0}$. If we denote by $\mathbb {E}_{\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }}$ the expectation of the probability measure $\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }$, we have that

$$\begin{aligned}&\mathbb {E}_{\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }}\tau _{B(\alpha ;\delta )}(X_N(\cdot ;x_N)) = \int \mathbb {E}\tau _{B(\alpha ;\delta )}(X_N(\cdot ;x)) \, \mu _N(\,\mathrm {d}x) \nonumber \\&\mathbb {E}_{\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }}\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) = \int \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \, \mu _\varepsilon (\,\mathrm {d}x). \end{aligned}$$

(7.15)

By Proposition 7.3$X_N(\cdot ;x_N)$ converges to $X(\cdot ;x)$$\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }$-almost surely on compact time intervals of $(0,\infty )$ up to a subsequence. Let

$$\begin{aligned} \mathtt {L} = \left\{ \delta \in (0,1/2): \mathbb {P}\left( \tau _{B(\alpha ;\delta )}(\cdot ) \text { is discontinuous on } X(\cdot ;x)\right) >0\right\} \end{aligned}$$

and notice that for $x(t) = L^{-2}\langle X(t;x), 1\rangle$

$$\begin{aligned}&\mathtt {L} \subset \left\{ \delta \in (0,1/2): \mathbb {P}\left( t\mapsto |x(t) - 1|\vee \Vert X(t;x) - x(t)\Vert _{\mathcal {C}^{-\alpha }} \right. \right. \\&\left. \left. \quad \qquad \text { has a local minimum at height } \delta \right) > 0 \right\} . \end{aligned}$$

As in [20, Proof of Theorem 6.1] the last set is at most countable, hence $\tau _{B(\alpha ;\delta )}(X_N(\cdot ;x_N)) \rightarrow \tau _{B(\alpha ;\delta )}(X(\cdot ;x))$$\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }$-almost surely up to a subsequence, except possibly a countable number of $\delta \in (0,1/2)$.

By Corollary 7.6 the family $\{\tau _{B(\alpha ;\delta )}(X_N(\cdot ;x))\}_{N\ge N_0}$ is uniformly integrable. Hence by Vitali’s convergence theorem (see [5, Theorem 4.5.4]) we obtain that

$$\begin{aligned} \mathbb {E}_{\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }}\tau _{B(\alpha ;\delta )}(X_N(\cdot ;x_N)) \rightarrow \mathbb {E}_{\mathbb {P}\otimes \mathbb {P}_{\mu _\varepsilon }}\tau _{B(\alpha ;\delta )}(X(\cdot ;x)). \end{aligned}$$

Combining with (7.5) and (7.15) the proof of the upper bound is complete. $\square$

7.3 Proof of Theorem 3.5

In this section we combine Proposition 7.7 and Theorem 3.1 to prove Theorem 3.5. The idea we use here was first implemented in the 1-dimensional case in [4]. Generally speaking, if we restrict ourselves on the event where the first transition from a neighbourhood of $-1$ to a neighbourhood of 1 happens after the exponential loss of memory, $\tau _{B(\alpha ;\delta )}(X(\cdot ;x))$ behaves like $\int \tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \, \mu _{\varepsilon }(\,\mathrm {d}x)$ for $x\in A(\alpha _0;\delta )$. The probability of this event is quantified by Theorem 3.1 and Proposition 7.8. On the complement of this event the transition time $\tau _{B(\alpha ;\delta )}(X(\cdot ;x))$ is estimated using Proposition 7.9.

In the next proposition we prove that the first transition from a neighbourhood of $-1$ to a neighbourhood of 1 happens only after some time $T_0>0$ with overwhelming probability. This is a large deviation event which can be estimated using continuity of X with respect to the initial condition x and the stochastic objects . We sketch the proof for completeness.

Proposition 7.8

For every $\alpha \in (0,\alpha _0)$ and $\delta \in (0,1/2)$ there exist $a_0,\delta _0,T_0>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{\Vert x-(-1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \le T_0) \le \mathrm {e}^{-a_0/\varepsilon }. \end{aligned}$$

Proof

We first notice that for $\Vert x-(-1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0$

$$\begin{aligned} \mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \ge T_0) \ge \mathbb {P}\left( \sup _{t\le T_0} \Vert X(t;x) - (-1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _1\right) , \end{aligned}$$

for some $\delta _1>0$. Using continuity of X with respect to x and the stochastic objects , the last probability can be estimated from below uniformly in $\Vert x-(-1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0$, for $\delta _0$ sufficiently small, by

for some $\delta _2>0$. Last by Proposition D.1 we find $a_0>0$ and $\varepsilon _0\in (0,1)$ such that for ever $\varepsilon \le \varepsilon _0$

which completes the proof. $\square$

In the next proposition we estimate the second moment of the transition time $\tau _{B(\alpha ;\delta )}(X(\cdot ;x))$ using the large deviation estimate (6.3). The proof combines the ideas in Propositions 6.4 and 6.6. However here we construct a path g which is different from the one in the proof of Proposition 6.4 to ensure that the process $X(\cdot ;x)$ returns to a neighbourhood of $-1$. The same proof implies exponential moments of the transition time $\tau _{B(\alpha ;\delta )}(X(\cdot ;x))$, but we only need to estimate the second moment in the proof of Theorem 3.5.

Proposition 7.9

Let $\alpha \in (0,\alpha _0)$ and $\delta \in (0,1/2)$. For every $\eta >0$ there exists $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;x))^2 \le C \mathrm {e}^{2\left[ (V(0)-V(-1))+\eta \right] /\varepsilon }, \end{aligned}$$

for some $C>0$ independent of $\eta$ and $\varepsilon$.

Proof

We first prove that for every $R,\eta >0$ there exists $T_0>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R}\mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x))\ge T_0) \le 1 - \mathrm {e}^{-\left[ (V(0)-V(-1))+ \eta \right] /\varepsilon }. \end{aligned}$$

We notice that there exists $\delta '>0$ such that

$$\begin{aligned} \mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x))\le T_0) \ge \mathbb {P}(\underbrace{\Vert X(T_*;x) - 1\Vert _{\mathcal {C}^{-\alpha }} \le \delta ' \text { for some } T_*\le T_0}_{=:\mathcal {A}(T_0;x)}). \end{aligned}$$

Here we use that if $\Vert X(T_*;x) - 1\Vert _{\mathcal {C}^{-\alpha }} \le \delta '$, for $\delta '$ sufficiently small then $X(T_*;x)\in B(\alpha ;\delta )$. By the large deviation estimate (6.3) we need to bound

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \sup _{\begin{array}{c} f \in \mathcal {A}(T_0;x) \\ f(0) = x \end{array}}I(f(\cdot ;x)). \end{aligned}$$

To do so we proceed as in the proof of Proposition 6.4 by constructing a suitable path $g\in \mathcal {A}(T_0;x)$. The construction here is similar but some of the steps differ since we need to ensure that g returns to a neighbourhood of 1. To avoid repeating ourselves we give a sketch of the proof highlighting the different steps of the construction.

Steps 1, 2 and 3 are exactly as in the proof of Proposition 6.4. However we need to distinguish the value of $\delta$ there from the value of $\delta$ in the statement of the proposition. If $g(\tau _3;x) \in B_{\mathcal {B}^1_{2,2}}(1;\delta )\cap B_{\mathcal {C}^{2+\lambda }}(0;C)$ we stop at Step 3. If not then $g(\tau _3;x) \in B_{\mathcal {B}^1_{2,2}}(-1;\delta )\cap B_{\mathcal {C}^{2+\lambda }}(0;C)$ or $B_{\mathcal {B}^1_{2,2}}(0;\delta )\cap B_{\mathcal {C}^{2+\lambda }}(0;C)$. We only explain how to proceed in the first case since it also covers the other.

Before we describe the remaining steps we recall that by Proposition C.2 there exist $y_{0,-},y_{0,+} \in B_{\mathcal {B}^1_{2,2}}(0;\delta )$ such that $y_{0,-},y_{0,+}\in \mathcal {C}^\infty$ and $X_{det}(t;y_{0,\pm }) \rightarrow \pm 1$ in $\mathcal {B}^1_{2,2}$. In particular there exists $T_0^*>0$ such that $X_{det}(T_0^*;y_{0,\pm })\in B_{\mathcal {B}^1_{2,2}}(\pm 1;\delta ) \cap B_{\mathcal {C}^{2+\lambda }}(0;C)$.

$\underline{\hbox {Step 4 (Jump to } X_{det}(T_0^*;y_{0,-})}$):

Let $\tau _4 = \tau _3+\tau$, for $\tau >0$ as in Step 2 which we fix below according to Lemma 6.5. For $t\in [\tau _3,\tau _4]$ we set $g(t;x) = g(\tau _3;x) + \frac{t - \tau _3}{\tau _4 - \tau _3}(X_{det}(T_0^*;y_{0,-}) - g(\tau _3;x))$.

Step 5 (Follow the deterministic flow backward to reach 0):

Let $\tau _5 = \tau _4+ T_0^*$. For $t\in [\tau _4,\tau _5]$ we set $g(t;x) = X_{det}(\tau _5 - t;y_{0,-})$.

$\underline{\hbox {Step 6 (Jump to } y_{0,+})}$:

Let $\tau _6 = \tau _5 +\tau$, for $\tau$ as in Step 4. For $t\in [\tau _5,\tau _6]$ we set $g(t;x) = g(\tau _5;x) + \frac{t - \tau _5}{\tau _6-\tau _5}(y_{0,+} - g(\tau _5;x))$.

Step 7 (Follow the deterministic flow forward to reach 1):

Let $\tau _7 = \tau _6+ T_0^*$. For $t\in [\tau _6,\tau _7]$ we set $g(t;x) = X_{det}(t - \tau _6;y_{0,+})$.

For the path g constructed above we notice that for every $\Vert x\Vert _{\mathcal {C}^{-\alpha _0}} \le R$, if $t\ge \tau _7$ then $g(t;x)\in B_{\mathcal {B}^1_{2,2}}(1;\delta )$. By (A.5), $\mathcal {B}^1_{2,2} \subset \mathcal {C}^{-\alpha }$, for every $\alpha >0$, hence if we choose $\delta$ sufficiently small and set $T_0 = \tau _7+1$ then $g\in \mathcal {A}(T_0;x)$.

To bound $I(g(\cdot ;x))$ we proceed exactly as in the proof of Proposition 6.4 using Lemma 6.5. But when considering the contribution from Step 5 we get

$$\begin{aligned}&\frac{1}{4}\int _{\tau _4}^{\tau _5} \Vert (\partial _t - \Delta ) g(t;x) + g(t;x)^3 - g(t;x)\Vert _{L^2}^2 \,\mathrm {d}t \\&\quad = 2 \int _0^{T_0^*} \left\langle \partial _t X_{det}(t;y_{0,+}), \Delta X_{det}(t;y_{0,+}) - X_{det}(t;y_{0,+})^3 + X_{det}(t;y_{0,+}) \right\rangle \,\mathrm {d}t \\&\quad = - 2 \left( V(X_{det}(T_0^*;y_0)) - V(y_{0,+})\right) \\&\quad \le 2 \left( V(0) - V(-1)\right) . \end{aligned}$$

In total we obtain the bound

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} I(g(\cdot ;x)) \le 2\left( V(0) - V(-1)\right) + C\delta . \end{aligned}$$

For $\eta >0$ we choose $\delta$ even smaller to ensure that $C\delta <\eta$. Then by (6.3) we find $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \inf _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \le T_0) \ge \mathrm {e}^{-\left[ (V(0) - V(-1)) + \eta \right] /\varepsilon }. \end{aligned}$$

The next step is to use the this estimate to show that for any $\eta >0$ there exists $\varepsilon _0\in (0,1)$ and possibly a different $T_0>0$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{x\in \mathcal {C}^{-\alpha _0}} \mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x)) \ge m T_0) \le \left( 1 - \mathrm {e}^{-\left[ (V(0) - V(-1)) + \eta \right] /\varepsilon }\right) ^m. \end{aligned}$$

We omit the proof since it is the same as the one of Proposition 6.6.

Finally we notice that

$$\begin{aligned} \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;x))^2&= \int _0^\infty 2t \, \mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x))\ge t) \,\mathrm {d}t \\&\le \sum _{m=0}^\infty \mathbb {P}(\tau _{B(\alpha ;\delta )}(X(\cdot ;x))\ge mT_0) \int _{mT_0}^{(m+1)T_0} 2t \,\mathrm {d}t \\&\le 2T_0^2 \sum _{m=0}^\infty (m+1) \left( 1 - \mathrm {e}^{-\left[ (V(0) - V(-1)) + \eta \right] /\varepsilon }\right) ^m \\&= 2 T_0^2 \mathrm {e}^{2\left[ (V(0) - V(-1)) + \eta \right] /\varepsilon }, \end{aligned}$$

which completes the proof. $\square$

Proof of Theorem 3.5

Let

$$\begin{aligned} \mathtt {Pr}(\varepsilon ) = \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} } \mathrm {e}^{\left( V(0) - V(-1)\right) /\varepsilon } \end{aligned}$$

and $\delta \in (0,\delta _0)$, for $\delta _0\in (0,1/2)$ which we fix below.

To prove the upper bound in (3.4) let $\delta _-<\delta$ and $T>0$ which we also fix below. For $x\in A(\alpha _0;\delta _-)$ we define the set

$$\begin{aligned}&A_T(x) = \left\{ \tau _{B(\alpha ;\delta _-)}(X(\cdot ;x)) > T, \sup _{\Vert \bar{y} - x\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \frac{\Vert X(t;{\bar{y}}) - X(t;x)\Vert _{\mathcal {C}^\beta }}{\Vert {\bar{y}}-x\Vert _{\mathcal {C}^{-\alpha _0}}} \right. \\&\left. \qquad \le C \mathrm {e}^{-(2-\kappa ) t} \text { for every } t\ge T\right\} , \end{aligned}$$

where $\delta _0$ and C are as in Theorem 3.1. For $y\in A(\alpha _0;\delta )$ and $x\in A(\alpha _0;\delta _-)$ we have that $\Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}, \Vert x-(-1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0$, if we choose $\delta _0$ sufficiently small. Furthermore for $y\in A(\alpha _0;\delta ), x\in A(\alpha _0;\delta _-)$ and $\omega \in A_T(x)$

$$\begin{aligned} \tau _{B(\alpha ;\delta )}(X(\cdot ;y)) \le \tau _{B(\alpha ;\delta _-)}(X(\cdot ;x)), \end{aligned}$$

if we choose T sufficiently large. By Proposition 7.8 and Theorem 3.1 there exist $a_1>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{x\in A(\alpha _0;\delta _-)} \mathbb {P}(A_T(x)^c) \le \sup _{\Vert x - (-1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \mathbb {P}(A_T(x)^c) \le \mathrm {e}^{-a_1/\varepsilon }. \end{aligned}$$

Then for every $y\in A(\alpha _0;\delta ), x\in A(\alpha _0;\delta _-)$ and $\eta >0$, which we fix below, there exists $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned}&\mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;y)) \nonumber \\&\quad \le \mathbb {E}\tau _{B(\alpha ;\delta _-)}(X(\cdot ;x)) + \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;y)){\mathbf {1}}_{A_T(x)^c} \nonumber \\&\quad {\mathop {\le }\limits ^{\text {Cauchy}-\text {Schwarz}}} \mathbb {E}\tau _{B(\alpha ;\delta _-)}(X(\cdot ;x)) + \left( \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;y))^2\right) ^{\frac{1}{2}} \mathbb {P}(A_T(x)^c)^{\frac{1}{2}} \nonumber \\&\quad {\mathop {\le }\limits ^{\text {Prop. }{}7.9}} \mathbb {E}\tau _{B(\alpha ;\delta _-)}(X(\cdot ;x)) + C \mathrm {e}^{\left( (V(0) - V(-1)) + \eta - \frac{a_1}{2}\right) /\varepsilon }, \end{aligned}$$

(7.16)

for some $C>0$ independent of $\varepsilon$. By Proposition 7.7 there exist $\delta _-\in (0,\delta ), c_+>0$ and $\mu _\varepsilon \in \mathcal {M}_1\left( A(\alpha _0;\delta _-)\right)$ such that for every $\varepsilon \in (0,1)$

$$\begin{aligned} \int \mathbb {E}\tau _{B(\alpha ;\delta _-)}(X(\cdot ;x)) \, \mu _\varepsilon (\,\mathrm {d}x) \le \mathtt {Pr}(\varepsilon ) (1+ c_+ \sqrt{\varepsilon }). \end{aligned}$$

Integrating (7.16) over x with respect to $\mu _\varepsilon$ implies that

$$\begin{aligned}&\sup _{y\in A(\alpha _0;\delta )} \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;y)) \\&\quad \le \mathtt {Pr}(\varepsilon ) \left( (1 + c_+\sqrt{\varepsilon }) + \mathrm {e}^{\left( \eta - \frac{a_1}{2}\right) /\varepsilon } C \left( \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} }\right) ^{-1} \right) . \end{aligned}$$

Let $\zeta >0$. Choosing $\eta <\frac{a_1}{2}$ we can find $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \mathrm {e}^{\left( \eta - \frac{a_1}{2}\right) /\varepsilon } C \left( \frac{2\pi }{|\lambda _0|} \sqrt{\prod _{k\in \mathbb {Z}^2}\frac{|\lambda _k|}{\nu _k} \exp \left\{ \frac{\nu _k-\lambda _k}{\lambda _k+2}\right\} }\right) ^{-1} \le \zeta \sqrt{\varepsilon }, \end{aligned}$$

which in turn implies that

$$\begin{aligned} \sup _{y\in A(\alpha _0;\delta )} \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;y)) \le \mathtt {Pr}(\varepsilon ) \left( 1 + (c_+ + \zeta )\sqrt{\varepsilon }\right) \end{aligned}$$

and proves the upper bound in (3.4).

To prove the lower bound, we let $\delta _+\in (\delta ,\delta _0)$ which we fix below and for $y\in A(\alpha _0;\delta )$ and $x\in A(\alpha _0;\delta _+)$ we define the set

$$\begin{aligned}&B_T(y,x) = \left\{ \tau _{B(\alpha ;\delta )}(X(\cdot ;y)) \ge T, \sup _{\Vert {\bar{y}}-x\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0} \frac{\Vert X(t;{\bar{y}}) - X(t;x)\Vert _{\mathcal {C}^\beta }}{\Vert {\bar{y}}-x\Vert _{\mathcal {C}^{-\alpha _0}}} \right. \\&\left. \qquad \qquad \qquad \le C \mathrm {e}^{-(2-\kappa ) t} \text { for every } t\ge T\right\} . \end{aligned}$$

For $y\in A(\alpha _0;\delta )$ and $x\in A(\alpha _0;\delta _+)$ we have that $\Vert y-x\Vert _{\mathcal {C}^{-\alpha _0}}, \Vert y-(-1)\Vert _{\mathcal {C}^{-\alpha _0}}, \Vert x-(-1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0$, if we choose $\delta _0$ sufficiently small. We also notice that for $y\in A(\alpha _0;\delta ), x\in A(\alpha _0;\delta _+)$ and $\omega \in B_T(y,x)$

$$\begin{aligned} \tau _{B(\alpha ;\delta _+)}(X(\cdot ;x)) \le \tau _{B(\alpha ;\delta )}(X(\cdot ;y)), \end{aligned}$$

if we choose T sufficiently large. By Proposition 7.8 and Theorem 3.1 there exists $a_1>0$ and $\varepsilon _0\in (0,1)$ such that for every $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \sup _{\begin{array}{c} y \in A(\alpha _0;\delta ) \\ x\in A(\alpha _0;\delta _+) \end{array}} \mathbb {P}(B_T(y,x)^c) \le \sup _{\begin{array}{c} \Vert y-(-1)\Vert _{\mathcal {C}^{-\alpha _0}}\le \delta _0 \\ \Vert x-(-1)\Vert _{\mathcal {C}^{-\alpha _0}} \le \delta _0 \end{array}} \mathbb {P}(B_T(y,x)^c) \le \mathrm {e}^{-a_1/\varepsilon }. \end{aligned}$$

Then for every $y\in A(\alpha _0;\delta ), x\in A(\alpha _0;\delta _+)$ and $\varepsilon \le \varepsilon _0$

$$\begin{aligned} \mathbb {E}\tau _{B(\alpha ;\delta )}(X(\cdot ;y))&\ge \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x)) {\mathbf {1}}_{B_T(y,x)} \\&= \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x)) - \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x)) {\mathbf {1}}_{B_T(y,x)^c} \\&{\mathop {\ge }\limits ^{\text {Cauchy}-\text {Schwarz}}} \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x)) \\&\quad - \left( \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x))^2\right) ^{\frac{1}{2}} \mathbb {P}\left( B_T(y,x)^c\right) ^{\frac{1}{2}} \\&\ge \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x)) - \left( \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x))^2\right) ^{\frac{1}{2}} \mathrm {e}^{-a_1/2\varepsilon } \end{aligned}$$

and we proceed as in the case of the upper bound, using Proposition 7.9 for $\mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x))^2$ and Proposition 7.7 to find $\delta _+\in (\delta ,\delta _0), c_->0$ and $\mu _\varepsilon \in \mathcal {M}_1\left( A(\alpha _0;\delta _+)\right)$ such that for every $\varepsilon \in (0,1)$

$$\begin{aligned} \int \mathbb {E}\tau _{B(\alpha ;\delta _+)}(X(\cdot ;x)) \, \mu _\varepsilon (\,\mathrm {d}x) \ge \mathtt {Pr}(\varepsilon ) (1 - c_- \varepsilon ). \end{aligned}$$

$\square$

Appendices

Besov spaces

Definition A.1

Let $\alpha \in \mathbb {R}$ and $p,q\in [1,\infty ]$. The Besov norm $\Vert \cdot \Vert _{\mathcal {B}^\alpha _{p,q}}$ is defined as

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}_{p,q}^\alpha }:=\left\| \left( 2^{\alpha \kappa }\Vert f*\eta _\kappa \Vert _{L^p}\right) _{\kappa \ge -1}\right\| _{\ell ^q}. \end{aligned}$$

(A.1)

Here the family of functions $\{\eta _\kappa \}_{\kappa \ge -1}$ is given by ${\hat{\eta }}_\kappa = \chi _\kappa$ in Fourier space for $\{\chi _\kappa \}_{\kappa \ge -1}$ a suitable dyadic partition of unity as in [1, Proposition 2.10]. The Besov space space $\mathcal {B}^\alpha _{p,q}$ is defined as the completion of $\mathcal {C}^\infty$ with respect to (A.1).

In this appendix we present several useful results from [19, 20] about Besov spaces that we repeatedly use in this article. For a complete survey of the full-space analogues of these results we refer the reader to [1]. A discussion on the validity of these results in the periodic case can be found in [20, Section 4.2].

The following estimate is immediate from the definition of the Besov norm (A.1),

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}^\alpha _{p,q}} \le C \Vert f\Vert _{\mathcal {B}^\beta _{p,q}}, \text { if } \beta > \alpha . \end{aligned}$$

(A.2)

Proposition A.2

([20, Remark 9]) Let $\alpha \in \mathbb {R}$ and $p,q_1,q_2\in [1,\infty ]$ such that $q_2> q_1$. For every $\lambda >0$

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}^\alpha _{p,q_2}} \le C \Vert f\Vert _{\mathcal {B}^{\alpha +\lambda }_{p,q_1}}. \end{aligned}$$

(A.3)

Proposition A.3

([20, Remarks 10 and 11]) For every $p\in [1,\infty ]$

$$\begin{aligned} C^{-1} \Vert f\Vert _{\mathcal {B}^0_{p,\infty }} \le \Vert f\Vert _{L^p} \le C \Vert f\Vert _{\mathcal {B}^0_{p,1}}. \end{aligned}$$

(A.4)

Proposition A.4

([20, Proposition 2]) Let $\alpha ,\beta \in \mathbb {R}$ and $p,q\ge 1$ such that $p\ge q$ and $\beta =\alpha + d \left( \frac{1}{q}-\frac{1}{p}\right)$. Then

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}_{p,\infty }^\alpha }\le C \Vert f\Vert _{\mathcal {B}_{q,\infty }^\beta }. \end{aligned}$$

(A.5)

Proposition A.5

([20, Proposition 5]) For every $\beta \ge \alpha$

$$\begin{aligned} \Vert \mathrm {e}^{t\Delta }f\Vert _{\mathcal {B}_{p,q}^\beta }\le C (t\wedge 1)^{\frac{\alpha -\beta }{2}}\Vert f\Vert _{\mathcal {B}_{p,q}^\alpha }. \end{aligned}$$

(A.6)

Proposition A.6

([20, Corollary 1]) Let $\alpha \ge 0$ and $p,q\in [1,\infty ]$. Then

$$\begin{aligned} \Vert fg\Vert _{\mathcal {B}^\alpha _{p,q}} \le C \Vert f\Vert _{\mathcal {B}^\alpha _{p_1,q_1}} \Vert g\Vert _{\mathcal {B}^\alpha _{p_2,q_2}}, \end{aligned}$$

(A.7)

where $p = \frac{1}{p_1}+\frac{1}{p_2}$ and $p = \frac{1}{q_1}+\frac{1}{q_2}$.

Proposition A.7

([20, Corollary 2]) Let $\alpha <0, \beta >0$ such that $\alpha +\beta >0$ and $p,q\in [1,\infty ]$. Then

$$\begin{aligned} \Vert fg\Vert _{\mathcal {B}^\alpha _{p,q}} \le C \Vert f\Vert _{\mathcal {B}^\alpha _{p_1,q_1}} \Vert g\Vert _{\mathcal {B}^\beta _{p_2,q_2}}, \end{aligned}$$

(A.8)

where $p = \frac{1}{p_1}+\frac{1}{p_2}$ and $p = \frac{1}{q_1}+\frac{1}{q_2}$.

Proposition A.8

([20, Proposition 10]) For every $\alpha <\alpha '$ the embedding $\mathcal {C}^{\alpha '}\hookrightarrow \mathcal {B}^\alpha _{\infty ,1}$ is compact.

Proposition A.9

([19, Proposition A.6]) For every $p\in [1,\infty )$

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}^1_{p,\infty }} \le C (\Vert \nabla f\Vert _{L^p} + \Vert f\Vert _{L^p}). \end{aligned}$$

Proposition A.10

([19, Corollary A.8]) Let $\alpha >0$ and $p,q\in [1,\infty ]$. Then

$$\begin{aligned} \Vert f^2\Vert _{\mathcal {B}^\alpha _{p,q}} \le C \Vert f\Vert _{L^{p_1}} \Vert f\Vert _{\mathcal {B}^\alpha _{p_2,q}}, \end{aligned}$$

(A.9)

where $p = \frac{1}{p_1}+\frac{1}{p_2}$.

In the next proposition we prove convergence of the Galerkin approximations $\Pi _Nf$ to f in Besov spaces. Here we use that the projection $\Pi _Nf$ is defined as the convolution of f with the 2-dimensional square Dirichlet kernel, which satisfies a logarithmic growth bound in the $L^1$ norm.

Proposition A.11

Let $\Pi _N: L^2 \rightarrow L^2$ be the projection on $\{f \in L^2: f(z)=\sum _{|k|\le N} {\hat{f}}(k) L^{-2}\mathrm {e}^{2\mathrm {i}\pi k\cdot z/L}\}$. Then for every $\alpha \in \mathbb {R}, p,q\in [1,\infty ]$ and $\lambda >0$

$$\begin{aligned}&\Vert \Pi _Nf -f\Vert _{\mathcal {B}^{\alpha }_{p,q}} \le \frac{C(\log N)^2}{N^\lambda } \Vert f\Vert _{\mathcal {B}^{\alpha +\lambda }_{p,q}} \end{aligned}$$

(A.10)

$$\begin{aligned}&\Vert \Pi _Nf\Vert _{\mathcal {B}^{\alpha }_{p,q}} \le C \Vert f\Vert _{\mathcal {B}^{\alpha +\lambda }_{p,q}}. \end{aligned}$$

(A.11)

If we furthermore assume that $p=2$ then

$$\begin{aligned}&\Vert \Pi _Nf -f\Vert _{\mathcal {B}^{\alpha }_{2,q}} \le \frac{C}{N^\lambda } \Vert f\Vert _{\mathcal {B}^{\alpha +\lambda }_{2,q}} \end{aligned}$$

(A.12)

$$\begin{aligned}&\Vert \Pi _Nf\Vert _{\mathcal {B}^{\alpha }_{2,q}} \le \Vert f\Vert _{\mathcal {B}^{\alpha }_{2,q}}. \end{aligned}$$

(A.13)

Proof

We first notice that for $c_2>c_1>0$

$$\begin{aligned} \delta _\kappa \left( \Pi _Nf - f\right) = {\left\{ \begin{array}{ll} 0, &{} \text { if } 2^\kappa \le c_1 N \\ \delta _\kappa f, &{} \text { if } 2^\kappa > c_2 N \end{array}\right. } . \end{aligned}$$

Let $D_N(z)=\sum _{|k|\le N} L^{-2} \mathrm {e}^{-2\mathrm {i}\pi k\cdot z/L}$ be the square Dirichlet kernel. Then $\Pi _N f = f*D_N$. Using the triangle inequality and Young’s inequality for convolution we have that

$$\begin{aligned} \Vert \delta _\kappa \left( \Pi _Nf - f\right) \Vert _{L^p} \le (\Vert D_N\Vert _{L^1}+1) \Vert \delta _\kappa f\Vert _{L^p}. \end{aligned}$$

Thus

$$\begin{aligned} \Vert \delta _\kappa (\Pi _Nf - f)\Vert _{L^p} \le {\left\{ \begin{array}{ll} 0,&{} \text { if } 2^\kappa \le c_1N \\ C (\log N)^2 \Vert \delta _\kappa f\Vert _{L^p}, &{} \text { if } c_1N \le 2^\kappa < c_2 N, \\ \Vert \delta _\kappa f\Vert _{L^p}, &{} \text { if } 2^\kappa > c_2 N \end{array}\right. } \end{aligned}$$

where in the second case we use that $\Vert D_N\Vert _{L^1}\lesssim (\log N)^2$. This bound immediate form the fact that the 2-dimensional square Dirichlet kernel is the product of two 1-dimensional Dirichlet kernels (see [11, Section 3.1.3]). The last implies (A.10) and (A.11). For $p=2$ we notice that

$$\begin{aligned} \Vert \delta _\kappa \Pi _Nf\Vert _{L^2} \le \Vert \delta _\kappa f\Vert _{L^2}, \end{aligned}$$

which implies (A.12) and (A.13). $\square$

Generalised Gronwall inequality

Lemma B.1

(Generalised Gronwall inequality) Let $f:[0,T] \rightarrow \mathbb {R}$ be a measurable function and $\sigma _1+\sigma _2<1$ such that

$$\begin{aligned} f(t) \le \mathrm {e}^{-c_0t} a + b \int _0^t \mathrm {e}^{-c_0(t-s)} (t-s)^{-\sigma _1} s^{-\sigma _2} f(s) \,\mathrm {d}s. \end{aligned}$$

Then there exists $c,C>0$ such that

$$\begin{aligned} f(t) \le C \exp \left\{ -c_0t + c b^{\frac{1}{1-\sigma _1-\sigma _2}} t\right\} a. \end{aligned}$$

Proof

The lemma is essentially [13, Lemma 5.7] if we set $x(t) = \mathrm {e}^{c_0t} f(t)$ with their notation. $\square$

Lemma B.2

Let $\alpha +\beta <1$ and $c>0$. Then

$$\begin{aligned} \sup _{t\ge 0} \int _0^t (t-s)^{-\alpha } (s\wedge 1)^{-\beta } \mathrm {e}^{-c(t-s)} \,\mathrm {d}s <\infty . \end{aligned}$$

Proof

Assume $t\ge 1$. Then

$$\begin{aligned} \int _0^1 (t-s)^{-\alpha } (s\wedge 1)^{-\beta } \mathrm {e}^{-c(t-s)} \,\mathrm {d}s \lesssim \mathrm {e}^{-ct} \int _0^t (t-s)^{-\alpha } (s\wedge 1)^{-\beta } \,\mathrm {d}s \lesssim t^{1-\alpha -\beta } \mathrm {e}^{-ct} \end{aligned}$$

and

$$\begin{aligned} \int _1^t (t-s)^{-\alpha } (s\wedge 1)^{-\beta } \mathrm {e}^{-c(t-s)} \,\mathrm {d}s&\le \int _0^t s^{-\alpha } \mathrm {e}^{-cs} \,\mathrm {d}s \lesssim 1 + \int _1^t s^{-\alpha } \mathrm {e}^{-cs} \,\mathrm {d}s \\&\lesssim 1+ \int _1^t \mathrm {e}^{-cs} \,\mathrm {d}s. \end{aligned}$$

The above implies that

$$\begin{aligned} \sup _{t\ge 1} \int _0^t (t-s)^{-\alpha } (s\wedge 1)^{-\beta } \mathrm {e}^{-c(t-s)} \,\mathrm {d}s <\infty . \end{aligned}$$

The bound for $t\le 1$ follows easily. $\square$

Deterministic dynamics

Propositions C.1 and C.2 are a consequence of [9, Section 8] and [16, Appendix B.1]. Although the results in [9, Section 8] concern 1 space-dimension they can be easily generalised in 2 space-dimensions. For consistency we have also replaced the space $H^1$ appearing in [9, Section 8] by $\mathcal {B}^1_{2,2}$. The fact that these spaces coincide is immediate from Definition A.1 for $p=q=2$ if we rewrite $\Vert f*\eta _k\Vert _{L^2}$ using Plancherel’s identity.

Proposition C.1

For every $x\in \mathcal {B}^1_{2,2}$ there exists $x_*\in \{-1,0,1\}$ such that $X_{det}(t;x) {\mathop {\rightarrow }\limits ^{\mathcal {B}^1_{2,2}}} x_*$.

Proposition C.2

For every $\delta >0$ there exists $x_\pm \in B_{\mathcal {B}^1_{2,2}}(0;\delta )$ such that $X_{det}(t;x_\pm ) {\mathop {\rightarrow }\limits ^{\mathcal {B}^1_{2,2}}} \pm 1$.

Proposition C.3

Let $R>0$. Then there exists $C\equiv C(R)>0$ such that for every $\lambda >0$ sufficiently small

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \Vert X_{det}(1;x)\Vert _{\mathcal {C}^{2+\lambda }} \le C. \end{aligned}$$

Proof

By [24, Theorem 3.3, Theorem 3.9] there exists $C\equiv C(R)>0$ such that

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R}\sup _{t\le 1} t^\gamma \Vert X_{\det }(t;x)\Vert _{\mathcal {C}^\beta } \le C. \end{aligned}$$

Let $S(t) = \mathrm {e}^{\Delta t}$. Using the mild form we write

$$\begin{aligned} X_{det}(1;x) = S(1/2)X_{det}\left( 1/2;x\right) - \int _{1/2}^1 S(1-s)\left( X_{det}(s;x)^3 + X_{det}(s;x)\right) \,\mathrm {d}s. \end{aligned}$$

Then

$$\begin{aligned}&\Vert X_{det}(1;x)\Vert _{\mathcal {C}^{2+\lambda }} \\&\quad \lesssim \Vert X_{det}\left( 1/2;x\right) \Vert _{\mathcal {C}^\beta } + \int _{1/2}^1 (1-s)^{-\frac{2+\lambda -\beta }{2}} \left( \Vert X_{det}(s;x)\Vert _{\mathcal {C}^\beta }^3 + \Vert X_{det}(s;x)\Vert _{\mathcal {C}^\beta }\right) \end{aligned}$$

and if we choose $\lambda <\beta$ the above implies that

$$\begin{aligned}&\sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R} \Vert X_{det}(1;x)\Vert _{\mathcal {C}^{2+\lambda }} \lesssim \sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R}\sup _{t\le 1} t^{3\gamma } \Vert X_{\det }(t;x)\Vert _{\mathcal {C}^\beta }^3 \\&\quad +\sup _{\Vert x\Vert _{\mathcal {C}^{-\alpha _0}}\le R}\sup _{t\le 1} t^\gamma \Vert X_{\det }(t;x)\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

$\square$

Stretched exponential moments for the stochastic objects

Proposition D.1

For every $n\ge 1$ there exists $c\equiv c(n)>0$ such that

Proof

Following step by step the proof of [24, Theorem 2.1] but using the explicit bound in Nelson’s estimate [24, Equation (B.3)] (see also [5, Section 1.6]), we have that for every $p\ge 1$

for some $C_n>0$. Then for any $c>0$

and by choosing $c\equiv c(n)>0$ sufficiently small the series converges. $\square$

An estimate for stochastically dominated random variables

Lemma E.1

Let $g_1,{\tilde{g}}_1$ be positive random variables such that

$$\begin{aligned} \mathbb {P}(g_1 \ge g) \le \mathbb {P}({\tilde{g}}_1 \ge g), \end{aligned}$$

for every $g\ge 0$ and let F be a positive decreasing measurable function on $[0,\infty )$. Then

$$\begin{aligned} \int _0^\infty F(g) \, \mu _{g_1}(\,\mathrm {d}g) \ge \int _0^\infty F(g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g), \end{aligned}$$

where $\mu _{g_1}$ and $\mu _{{\tilde{g}}_1}$ is the law of $g_1$ and ${\tilde{g}}_1$.

Proof

We first assume that F is smooth. Then $\frac{\,\mathrm {d}}{\,\mathrm {d}g}F(g) \le 0$ for every $g\ge 0$. Hence

$$\begin{aligned} \int _0^\infty F(g) \, \mu _{g_1}(\,\mathrm {d}g)&= F(0) + \int _0^\infty \frac{\,\mathrm {d}}{\,\mathrm {d}g}F(g) \, \mathbb {P}(g_1 \ge g) \,\mathrm {d}g \ge F(0) \\&\quad + \int _0^\infty \frac{\,\mathrm {d}}{\,\mathrm {d}g}F(g) \, \mathbb {P}({\tilde{g}}_1 \ge g) \,\mathrm {d}g \\&= \int _0^\infty F(g) \, \mu _{{\tilde{g}}_1}(\,\mathrm {d}g), \end{aligned}$$

which proves the estimate for F differentiable. To prove the estimate for a general decreasing function F we define $F_\delta = F*\eta _\delta$ for some positive mollifier $\eta _\delta$ to preserve monotonicity and use the last estimate together with the dominated convergence theorem. $\square$

Proof of Proposition 7.3

Proof of Proposition 7.3

By [24, Proposition 2.3] for every $\alpha >0, p\ge 1$ and $T>0$

Hence convergences to 0 in probability.

It is enough to prove that

$$\begin{aligned} \lim _{N\rightarrow \infty } \sup _{x\in \aleph } \sup _{t\le T} (t\wedge 1)^\gamma \Vert v_N(t;x_N) - v(t;x)\Vert _{\mathcal {C}^\beta } = 0. \end{aligned}$$

This, convergence in probability of to 0 and the embedding $\mathcal {C}^\beta \subset \mathcal {C}^{-\alpha }$ (see (A.2)) imply the result.

Let $S(t) = \mathrm {e}^{\Delta t}$. For simplicity we write $v_N(t)$ and v(t) to denote $v_N(t;x_N)$ and v(t; x). Using the mild forms of (7.2) and (2.2) we get

(F.1)

Let $\iota =\inf \{t>0:(t\wedge 1)^\gamma \Vert v_N(t) - v(t)\Vert _{\mathcal {C}^\beta }\ge 1\}$ and $t\le T\wedge \iota$. We treat each of the terms in (F.1) separately. Below the parameters $\alpha$ and $\lambda$ can be taken arbitrarily small and all the implicit constants depend on , and $\sup _{x\in \aleph }\sup _{t\le T}(t\wedge 1)^\gamma \Vert v(t)\Vert _{\mathcal {C}^\beta }$.

Term ${I_1}$:

$$\begin{aligned} I_1 {\mathop {\lesssim }\limits ^{(\mathrm{A.6})}} (t\wedge 1)^{-\frac{\alpha _0+\beta }{2}} \sup _{x\in \aleph } \Vert x_N - x\Vert _{\mathcal {C}^{-\alpha _0}} \end{aligned}$$

Term ${I_2}$:

$$\begin{aligned} I_2&{\mathop {\lesssim }\limits ^{(\mathrm{A.6})}} \int _0^t \left( (t-s)^{-\frac{\lambda }{2}} \Vert \Pi _N(v_N(s)^3) - v_N(s)^3\Vert _{\mathcal {C}^{\beta -\lambda }} + \Vert v_N(s)^3 - v(s)^3\Vert _{\mathcal {C}^\beta }\right) \,\mathrm {d}s\\&{\mathop {\lesssim }\limits ^{(\mathrm{A.10})}} \int _0^t (t-s)^{-\frac{\lambda }{2}} \left( \frac{(\log N)^2}{N^\lambda } \Vert v_N(s)^3\Vert _{\mathcal {C}^\beta } + \Vert v_N(s)^3 - v(s)^3\Vert _{\mathcal {C}^{\beta -\lambda }} \right) \,\mathrm {d}s \\&{\mathop {\lesssim }\limits ^{(\mathrm{A.7})}} \int _0^t \bigg [(t-s)^{-\frac{\lambda }{2}} \frac{(\log N)^2}{N^\lambda } \Vert v_N(s)\Vert _{\mathcal {C}^\beta }^3 + \Vert v_N(s) - v(s)\Vert _{\mathcal {C}^\beta } \\&\quad \times \left( \Vert v_N(s)\Vert _{\mathcal {C}^\beta }^2 + \Vert v_N(s)\Vert _{\mathcal {C}^\beta } \Vert v(s)\Vert _{\mathcal {C}^\beta } + \Vert v(s)\Vert _{\mathcal {C}^\beta }^2\right) \bigg ] \,\mathrm {d}s \\&\lesssim \int _0^t \left( (t-s)^{-\frac{\beta +\frac{2}{p}-1}{2}} \frac{(\log N)^2}{N^\lambda } (s\wedge 1)^{-3\gamma } + (s\wedge 1)^{-2\gamma } \Vert v_N(s) - v(s)\Vert _{\mathcal {C}^\beta } \right) \,\mathrm {d}s. \end{aligned}$$

Term ${I_3}$:

Term ${I_4}$: Similarly to $I_3$,

Term ${I_5}$:

Terms ${I_6, I_7}$:

Combining the above estimates we obtain that for $t\le T\wedge \iota$

By the generalised Gronwall inequality, Lemma B.1, on $f(t) = (t\wedge 1)^\gamma \Vert v_N(t) - v(t)\Vert _{\mathcal {C}^\beta }$ we find $C\equiv C(T)>0$ such that

This and convergence of to 0 in probability imply the result. $\square$

Local existence in $L^2$-based Besov spaces

In this section we fix $\beta \in \left( \frac{1}{3},\frac{2}{3}\right) , \gamma \in \left( \frac{\beta }{2},\frac{1}{3}\right)$ and $p\in (1,2)$ such that

$$\begin{aligned} 1-\frac{2}{3p} <\beta \text { and } 1-\frac{\beta +\frac{2}{p}-1}{2}-2\gamma > 0. \end{aligned}$$

The next proposition provides local existence of (7.2) in $\mathcal {B}^\beta _{2,2}$ up to some time $T_*>0$ which is uniform in the regularisation parameter N.

Proposition G.1

Let $K, R, T>0$ such that $\Vert x\Vert _{\mathcal {B}^{-\alpha _0}_{2,2}} \le R$ and . Then there exist $T_*\equiv T_*(K,R) \le T$ and $C\equiv C(K,R)>0$ such that (7.2) has a unique solution $v\in C((0,T_*];\mathcal {B}^\beta _{2,2})$ satisfying

$$\begin{aligned} \sup _{t\le T_*} (t\wedge 1)^\gamma \Vert v_N(t;x)\Vert _{\mathcal {B}^\beta _{2,2}} \le C. \end{aligned}$$

Proof

Let $S(t) = \mathrm {e}^{\Delta t}$. We define

It is enough to prove that there exists $T_*>0$ such that ${\mathscr {T}}$ is a contraction on

$$\begin{aligned} {\mathscr {B}}_{T_*} := \left\{ v: \sup _{t\le T_*}(t\wedge 1)^\gamma \Vert v(t;x)\Vert _{\mathcal {B}^\beta _{2,2}} \le 1\right\} . \end{aligned}$$

We first prove that for $T_*>0$ sufficiently small ${\mathscr {T}}$ maps ${\mathscr {B}}_{T_*}$ to itself. To do so we notice that

where we use (A.13) together with the relation $S(\cdot )\Pi _N = \Pi _NS(\cdot )$ to drop $\Pi _N$. We treat each term separately.

Term ${I_1}$:

$$\begin{aligned} I_1 {\mathop {\lesssim }\limits ^{(\mathrm{A.6})}} (t\wedge 1)^{-\frac{\alpha _0+\beta }{2}} \Vert x\Vert _{\mathcal {B}^{-\alpha _0}_{2,2}} \lesssim (t\wedge 1)^{-\frac{\alpha _0+\beta }{2}} R. \end{aligned}$$

Term ${I_2}$:

$$\begin{aligned} I_2&{\mathop {\lesssim }\limits ^{(\mathrm{A.5})}} \int _0^t \Vert S(t-s)v(s)^3\Vert _{\mathcal {B}^{\beta +\frac{2}{p}-1}_{p,2}} \,\mathrm {d}s {\mathop {\lesssim }\limits ^{(\mathrm{A.6})}} \int _0^t (t-s)^{-\frac{\beta +\frac{2}{p}-1}{2}} \Vert v(s)^3\Vert _{\mathcal {B}^0_{p,2}} \,\mathrm {d}s \\&{\mathop {\lesssim }\limits ^{(\mathrm{A.7})}} \int _0^t (t-s)^{-\frac{\beta +\frac{2}{p}-1}{2}} \Vert v(s)\Vert _{\mathcal {B}^0_{3p,2}}^3 \,\mathrm {d}s {\mathop {\lesssim }\limits ^{(\mathrm{A.5})}} \int _0^t (t-s)^{-\frac{\beta +\frac{2}{p}-1}{2}} \Vert v(s)\Vert _{\mathcal {B}^{1-\frac{2}{3p}}_{2,2}}^3 \,\mathrm {d}s \\&{\mathop {\lesssim }\limits ^{1-\frac{2}{3p}<\beta }} \int _0^t (t-s)^{-\frac{\beta +\frac{2}{p}-1}{2}} \Vert v(s)\Vert _{\mathcal {B}^\beta _{2,2}}^3 \,\mathrm {d}s \lesssim \int _0^t (t-s)^{-\frac{\beta +\frac{2}{p}-1}{2}} (s\wedge 1)^{-3\gamma } \,\mathrm {d}s. \end{aligned}$$

Term ${I_3}$:

Term ${I_4}$:

Terms ${I_5, I_6, I_7}$:

Combining all the above we find $C\equiv C(K,R)>0$ such that

$$\begin{aligned} \sup _{t\le T_*} (t\wedge 1)^\gamma \Vert {\mathscr {T}}(v)(t)\Vert _{\mathcal {B}^\beta _{2,2}} \le C T_*^\theta , \end{aligned}$$

for some $\theta \equiv \theta (\alpha ,\alpha ',\alpha _0,\beta ,\gamma )\in (0,1)$. Choosing $T_*>0$ sufficiently small the above implies that

$$\begin{aligned} \sup _{t\le T_*} (t\wedge 1)^\gamma \Vert {\mathscr {T}}(v)(t)\Vert _{\mathcal {B}^\beta _{2,2}} \le 1. \end{aligned}$$

Hence for this choice of $T_*, {\mathscr {T}}$ maps ${\mathscr {B}}_{T_*}$ to itself. In a similar way, but by possibly choosing a smaller value of $T_*$, we prove that ${\mathscr {T}}$ is a contraction on ${\mathscr {B}}_{T_*}$. For simplicity we omit the proof. That way we obtain a unique solution $v\in C((0,T_*];\mathcal {B}^\beta _{2,2})$. We can furthermore assume that $T_*$ is maximal in the sense that either $T_* = T$ or $\lim _{t\nearrow T_*} \Vert v(t;x)\Vert _{\mathcal {B}^\beta _{2,2}} = \infty$. $\square$

Proposition G.2

For every $t_0\in (0,1)$, and $K,R>0$ there exists $C\equiv C(t_0,K,R)>0$ such that if $\Vert x\Vert _{\mathcal {B}^{-\alpha }_{2,2}} \le R$ and then

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {B}^{-\alpha }_{2,2}} \le R} \Vert X_N(t_0;x)\Vert _{\mathcal {C}^{-\alpha }} \le C. \end{aligned}$$

Proof

Using the a priori estimate in Proposition 7.4 we can assume that $T_* =1$ in Proposition G.1. This implies that

$$\begin{aligned} \sup _{\Vert x\Vert _{\mathcal {B}^{-\alpha }_{2,2}}\le R}\sup _{t\le 1} t^\gamma \Vert v_N(t;x)\Vert _{\mathcal {B}^\beta _{2,2}} \le C. \end{aligned}$$

(G.1)

For simplicity we assume that $t_0=1$. Let $S(t) = \mathrm {e}^{\Delta t}$. Using the mild form of (7.2) we obtain that

We treat each term separately.

Term ${I_1}$:

$$\begin{aligned} I_1 {\mathop {\lesssim }\limits ^{(\mathrm{A.5})}} \Vert S(1/2)v_N(1/2)\Vert _{\mathcal {B}^{-\alpha +1}_{2,\infty }} {\mathop {\lesssim }\limits ^{(\mathrm{A.6})}} \Vert v_N(1/2)\Vert _{\mathcal {B}^{-\alpha }_{2,\infty }} \lesssim \Vert v_N(1/2)\Vert _{\mathcal {B}^{-\alpha }_{2,2}} \end{aligned}$$

Term ${I_2}$:

$$\begin{aligned} I_2&{\mathop {\lesssim }\limits ^{(\mathrm{A.5})}} \int _{1/2}^1 \Vert S(1-s)\Pi _N(v_N(s)^3)\Vert _{\mathcal {B}^{-\alpha +\frac{2}{p}}_{p,\infty }} \,\mathrm {d}s\\&{\mathop {\lesssim }\limits ^{(\mathrm{A.6})}} \int _{1/2}^1 (1-s)^{-\frac{-\alpha +\frac{2}{p}+\lambda }{2}} \Vert \Pi _N(v_N(s)^3)\Vert _{\mathcal {B}^{-\lambda }_{p,\infty }} \,\mathrm {d}s \\&{\mathop {\lesssim }\limits ^{(\mathrm{A.11}), (\mathrm{A.7})}} \int _{1/2}^1 (1-s)^{-\frac{-\alpha +\frac{2}{p}+\lambda }{2}} \Vert v_N(s)\Vert _{\mathcal {B}^0_{3p,\infty }}^3 \,\mathrm {d}s\\&{\mathop {\lesssim }\limits ^{(\mathrm{A.5})}} \int _{1/2}^1 (1-s)^{-\frac{-\alpha +\frac{2}{p}+\lambda }{2}} \Vert v_N(s)\Vert _{\mathcal {B}^{1-\frac{2}{3p}}_{2,\infty }}^3 \,\mathrm {d}s \\&{\mathop {\lesssim }\limits ^{1-\frac{2}{3p}<\beta }} \int _{1/2}^1 (1-s)^{-\frac{-\alpha +\frac{2}{p}+\lambda }{2}} \Vert v_N(s)\Vert _{\mathcal {B}^\beta _{2,2}}^3 \,\mathrm {d}s. \end{aligned}$$

Term ${I_3}$:

Term ${I_4}$: Similarly to $I_3$,

Terms ${I_5, I_6, I_7}$:

The proof is complete if we combine these estimates with (G.1). $\square$