A stochastic model of chemorepulsion with additive noise and nonlinear sensitivity

Abstract

We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By establishing an a priori estimate independent of the initial data, we show that there exists a pathwise unique, global solution to the SPDE. Furthermore, we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder–Besov space of positive regularity, which the solution law converges to exponentially fast. The a priori bound also allows us to establish tail estimates on the $L^p$ norm of the invariant measure which are heavier than Gaussian.

1 Introduction

We study the Cauchy problem and establish exponential ergodicity for the SPDE

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u -\partial _{xx} u = \chi \partial _x (u^2\partial _x \rho _u)+\xi , &{}\text { on }\mathbb {R}_+\times \mathbb {T},\\ -\partial _{xx}\rho _u = u-\bar{u}, &{}\text { on } \mathbb {T},\\ u|_{t=0}= \zeta , &{}\text { on }\mathbb {T}, \end{array}\right. } \end{aligned}$$

(1.1)

where $\xi$ is a space-time white noise, $\mathbb {T}:= \mathbb {R}/\mathbb {Z}$ is the one dimensional torus with unit volume and $\chi >0$ is a positive constant. The spatial average, $\bar{u}:=\langle u,1\rangle _{L^2(\mathbb {T})}$, is equal to the $0^{\text {th}}$ Fourier mode and we will take $\zeta$ to be a Hölder–Besov distribution of possibly negative regularity. Without loss of generality we assume that the solution to the second equation of (1.1) is mean free.

Without noise, (1.1) is a Keller–Segel model of chemorepulsion with nonlinear sensitivity and is an example from a wide family of parabolic PDE models of chemotaxis written, with some generality on the d-dimensional torus, $\mathbb {T}^d$, for $a\ge 0$, as

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - \Delta u = \nabla \cdot (f(u,\rho )\nabla \rho ), &{} \text { on } \mathbb {R}_+\times \mathbb {T}^d,\\ a \partial _t \rho - \Delta \rho = u - \bar{u},&{}\text { on }\mathbb {R}_+\times \mathbb {T}^d,\\ u\big |_{t=0}= u_0\ge 0,\quad \rho \big |_{t=0}=\rho _0, &{} \text { on }\mathbb {T}^d. \end{array}\right. } \end{aligned}$$

(1.2)

Chemotaxis refers to the directed movement of cells or bacteria in response to a chemical field. In (1.2), u represents the cell or bacteria density, $\rho$ the strength of the chemical field and $f\in C^\infty (\mathbb {R}^2;\mathbb {R})$ is a density dependent sensitivity function such that $f(0,y)=0$ for all $y\in \mathbb {R}$. The viscous terms are included to account for microscopic fluctuations in the cell and chemical diffusions. Chemorepulsion refers to a situation where cells are biased to move down the chemical gradient, modelled by choosing $f(x,y)\ge 0$ for $x\ge 0$, while chemoattraction refers to the situation where cells are biased to move up the gradient, modelled by choosing $f(x,y)\le 0$ for $x\ge 0$. In situations of typical interest, the chemical field will be generated by the cell population itself, which leads to the nonlinearity in (1.2). When $f(u,\rho )=\chi u$, for $\chi \in \mathbb {R}$, (1.2) is referred to as the Patlak–Keller–Segel model and was first introduced by Keller and Segel in [15], building on earlier work by Patlak, [28]. When $a =0$, (1.2) is known as a parabolic-elliptic model and when $a>0$, as a parabolic-parabolic model. A common feature of chemotaxis models is strongly differing behaviour in the chemorepulsive and chemoattractive settings. For example, in the parabolic-elliptic Patlak–Keller–Segel model ($f(u,\rho )=\chi u$, $a=0$), global well-posedness holds in all dimensions in the chemorepulsive regime ($\chi >0$) and in both regimes (any $\chi \in \mathbb {R}$) when $d=1$. On the other hand, if $d\ge 2$, in the chemoattractive regime ($\chi <0$) there exists a finite threshold $-\infty<\bar{\chi }_d<0$ such that, under some additional conditions when $d>2$, the model exhibits finite time blow-up for all $\chi <\bar{\chi }_d$. We refer to [11, 29] for a more detailed exposition of these phenomena and discussion of other models. For the one dimensional setting in particular see [12, 25] and for the chemorepulsive setting see [33].

In this paper we study (1.1), a one-dimensional additive noise, stochastic version of (1.2) in the chemorepulsive and parabolic–elliptic ($a=0$) regime, with nonlinear sensitivity function, $f(u,\rho )=\chi u^2$. Deterministic models including nonlinear sensitivity have been studied since the early work of Keller and Segel, see for example [7, 16, 18, 26]. These works focused on modelling nonlinear sensitivity with respect to the chemical signal, i.e. $f(u,\rho )= \chi u \tilde{f}(\rho )$. More recently there has been interest in modelling nonlinear sensitivity with respect to the cell density, i.e. $f(u,\rho )= \tilde{f}(u)$. This interest has been motivated, in part, by desires to include volume filling, saturation and density dependent (quorum sensing) regularisation effects into chemotaxis models, [9, 10, 13, 17, 31, 32]. The nonlinearity that we treat in (1.1) was already considered in the deterministic context in [17, 31]; in [17] it was shown that in the chemorepulsive setting the parabolic-elliptic model exhibits global existence and convergence to equilibrium when $\tilde{f}(u)\sim u^m$ for all $m>0$. In this case, the nonlinear sensitivity models an increased response to the chemorepulsant when cell density is high.

While it is common to study chemotaxis using PDE models, SPDE approaches can be used to model exogenous processes, study meta-stability or replicate physically relevant fluctuations around the continuum limit. In this work we focus on an additive noise model for two reasons. Firstly, it is a relatively simple model for which we can accommodate space-time white noise and establish a full ergodic analysis. Secondly, it allows us to make meaningful comparisons with additive noise stochastic reaction-diffusion equations which have been extensively studied; we refer to [4, 20, 23, 27, 34] for an incomplete list of works concerning these models. In particular, we highlight a comparison between our stochastic model, (1.1), and those considered in [20]. As in the above works, the main technical step we require is to obtain sufficiently strong a priori bounds on the pathwise solution which are independent of the initial data. In our context we divide the solution into a regular and an irregular part and the key step is a careful analysis of the more regular, remainder equation exploiting the repulsive nature of our nonlinearity to compensate for lower order terms without definitive sign. This is the content of Theorem 4.4. Furthermore, we keep careful track of the exponents and constants through the proof, which in particular allows us to obtain a tail bound, which is heavier than Gaussian, on the $L^p$ norm of the invariant measure. We give a detailed discussion on the implications of this bound and comparisons to the literature on stochastic reaction-diffusion equations in Remarks 4.5–4.7.

There are many natural questions that arise from our work. For this model it would be interesting to understand if the heavier than Gaussian tails that we establish are in fact optimal. Furthermore, applying our current methods, it appears that the low integrability of the solution, as compared with stochastic reaction-diffusion equations, imposes a significant barrier to considering $m>2$ in (1.1). Understanding whether this is a genuine issue or simply one of methodology would be interesting. Going further, one could naturally ask if it is possible to extend the well-known global well-posedness of attractive chemotaxis models in one dimension, [12, 25], to the stochastic case with white noise forcing. In addition, since chemotaxis is typically observed in two or three dimensions, extending the above study to higher dimensional versions would certainly be of interest. Using simple power counting one would expect (1.1) to be sub-critical in the sense of regularity structures, [8], on $\mathbb {R}_+\times \mathbb {T}^d$ for $d<4$. While local analysis is tractable in the presence of white-noise forcing, establishing global well-posedness for the repulsive model in dimension two seems challenging by our current methods, see [19, Ch. 7]. Finally, beyond the additive noise case, both multiplicative and conservative noise models are highly relevant in the context of chemotaxis. Extending our results to those cases, especially in a two or three dimensional setting, is a natural goal.

In the remainder of the introduction we summarise some notation in Sect. 1.1, and present our main results in Sect. 1.2. In Sect. 2 we recap the definitions and properties of the linear stochastic heat equation, which plays a key role in our analysis. In Sect. 3 we obtain local well-posedness of (1.1) and some ancillary properties of the solution. Sect. 4 contains the main contribution of this paper, establishing an a priori bound that is independent of the initial data and constitutes a coming down from infinity property. This a priori bound enables us to establish global well-posedness, existence of invariant measures and the tail bound, (1.5), below. In Sect. 5 we establish the existence of invariant measures to (1.1) and closely following the approach of [34], we establish the strong Feller property, irreducibility, and exponential ergodicity of the associated semi-group. The “Appendices, A and B” respectively contain summaries and useful properties of the inhomogeneous Hölder–Besov spaces and the stochastic heat equation on $\mathbb {T}$.

1.1 Notation

Let $\mathbb {N}=\{0,1,\ldots ,\}$ denote the set of non-negative integers. For $k\in \mathbb {N}$, we denote by $C^k(\mathbb {T})$ the space of k times continuously differentiable, 1-periodic, real functions. We denote by $C^\infty (\mathbb {T})$ the space of smooth, periodic functions with values in $\mathbb {R}$ and $\mathcal {S}'(\mathbb {T})$ for its dual. For $k\in \mathbb {N},\,p\in [1,\infty )$ (resp. $p=\infty$) we write $W^{k,p}(\mathbb {T})$ for the spaces of periodic functions with p-integrable (resp. essentially bounded) weak derivatives up to $k^{\text {th}}$ order, and write $L^p(\mathbb {T})=W^{0,p}(\mathbb {T})$. We write $\mathcal {B}^\alpha _{p,q}(\mathbb {T})$ for the Besov space associated to $\alpha \in \mathbb {R}$, $p,q\in [1,\infty ]$ and use the shorthand $\mathcal {C}^\alpha (\mathbb {T})=\mathcal {B}^\alpha _{\infty ,\infty }(\mathbb {T})$. Note that for $\alpha \in \mathbb {N}$ the spaces $\mathcal {C}^\alpha$ and $C^\alpha$ are not equal. See “Appendix A” for a full definition.

For $f \in \mathcal {S}'(\mathbb {T})$ we denote its spatial mean by $\bar{f}:=\langle 1,f\rangle _{L^2(\mathbb {T})}$. For $\mathfrak {m}\in \mathbb {R}$ we write, for example, $\mathcal {S}'_\mathfrak {m}(\mathbb {T}),C^k_\mathfrak {m}(\mathbb {T}), \mathcal {C}^\alpha _\mathfrak {m}(\mathbb {T}), L^p_{\mathfrak {m}}(\mathbb {T})$, for the corresponding spaces with the additional constraint that $\bar{f} =\mathfrak {m}$. When the context is clear we drop dependence on the domain in order to lighten notation.

Given a Banach space E, a subset $I\subseteq [0,\infty )$ and $\kappa \in (0,1)$, we write $C_IE:= C(I;E)$ (resp. $\mathcal {C}^{\kappa }_I E := \mathcal {C}^{\kappa }(I;E)$) for the space of continuous (resp. $\kappa$-Hölder continuous) maps $f:I\rightarrow E$ equipped with the norm $\Vert f\Vert _{C_IE}:= \sup _{t\in I} \Vert f_t\Vert _E$ (resp. $\Vert f\Vert _{\mathcal {C}_I^{\kappa }E}:=\Vert f\Vert _{C_IE}+ \sup _{t \ne s \in I} \frac{\Vert f_t-f_s\Vert _{E}}{|t-s|^{\kappa }}$). For $T>0$, we use the shorthand $C_TE=C_{[0,T]}E$ and $\mathcal {C}_T^\kappa E=\mathcal {C}_{[0,T]}^\kappa E$. Note that the norm $\Vert f\Vert _{\mathcal {C}_T^{\kappa }E}$ is equivalent to $\Vert f_0\Vert _E + \sup _{t\ne s \in [0,T]}\frac{\Vert f_t-f_s\Vert _E}{|t-s|^\kappa }$. For $\eta >0$ we let $C_{\eta ;T}E:= C_\eta ((0,T];E)$ be the set of continuous functions $f:(0,T]\rightarrow E$ such that

$$\begin{aligned} \Vert f\Vert _{C_{\eta ;T}E} := \sup _{ t \in (0,T]} (t^\eta \wedge 1)\Vert f_t\Vert _E < \infty . \end{aligned}$$

For a Banach space, E, equipped with its Borel $\sigma$-algebra, $\mathcal {E}$, we use the notation $\mathcal {B}_b(E)$, $\mathcal {C}_b(E)$ and $\mathcal {C}^1_b(E)$ respectively for the sets of bounded Borel measurable, continuous and continuously Fréchet differentiable maps $\Phi :E\rightarrow \mathbb {R}$. We write $\mathcal {P}(E)$ for the set of probability measures on E which we equip with the topology of weak convergence. For a sequence $(\mu _n)_{n\ge 1} \subset \mathcal {P}(E)$ we write $\mu _n \rightharpoonup \mu$ to indicate the weak convergence of $(\mu _n)_{n\ge 1}$ to $\mu \in \mathcal {P}(E)$. Using $\Pi _{\mu ,\nu }$ to denote the set of all couplings between $\mu ,\,\nu \in \mathcal {P}(E)$, we write the total variation distance as,

$$\begin{aligned} \Vert \mu -\nu \Vert _{\mathrm {TV}} := \sup _{A \in \mathcal {E}}|\mu (A)-\nu (A)| = \inf _{\pi \in \Pi _{\mu ,\nu }} \iint _{E^2} \mathbbm {1}_{x=y}\mathop {}\!\mathrm {d}\pi (x,y). \end{aligned}$$

We write $\lesssim$ to indicate that an inequality holds up to a constant depending on quantities that we do not keep track of or are fixed throughout. When we do wish to emphasise the dependence on certain quantities, we either write $\lesssim _{K,v}$ or define $C:=C(\alpha ,p,d)>0$ and write $\le C$.

1.2 Main results

We fix a filtered probability space $(\Omega ,\mathcal {F},(\mathcal {F}_t)_{t\ge 0},\mathbb {P})$ carrying a mean-free, space-time white noise, $\xi$, defined in Sect. 2 below. We fix $\alpha _0\in \left( -\frac{1}{2},0\right)$, $\alpha \in \left( 0,\alpha _0+\frac{1}{2}\right)$ and $\eta >0$ such that,

$$\begin{aligned} \frac{\alpha -\alpha _0}{2}<\eta < \frac{1}{4}. \end{aligned}$$

(1.3)

Theorem 1.1

(Global Well-Posedness) Let $T>0$, $\mathfrak {m}\in \mathbb {R}$, $\chi >0$ and $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$. Then there exists a unique, probabilistically strong, mild solution, $u(\zeta )$, to (1.1) such that $\mathbb {P}$-a.s. $u(\zeta )\in C_{\eta ;T}\mathcal {C}_\mathfrak {m}^{\alpha }(\mathbb {T})$.

Remark 1.2

While we frame Theorem 1.1 as a probabilistic statement, the proof is mainly deterministic and based on PDE techniques. In the course of the proof we additionally establish local Lipschitz continuity of the solution in both initial data and the noise, see Proposition 4.8. This will be important in Sect. 5.3 when we establish full support of the law on $\mathcal {C}^{\alpha }_{\mathfrak {m}}(\mathbb {T})$.

Remark 1.3

The statement of Theorem 1.1 remains valid for $\eta =\frac{\alpha -\alpha _0}{2}$. However, we take $\eta >\frac{\alpha -\alpha _0}{2}$ to simplify some statements and proofs below. In general we may think of $\eta$ as being arbitrarily close to $\frac{\alpha -\alpha _0}{2}$.

Let $\mathcal {L}(u_t(\zeta )) := u_t(\zeta )\# \mathbb {P}\in \mathcal {P}\big (\mathcal {C}_{\mathfrak {m}}^{\alpha }(\mathbb {T})\big )$ denote the law of the solution, started from $\zeta \in \mathcal {C}^{\alpha _0}_{\mathfrak {m}}(\mathbb {T})$, at time $t>0$.

Theorem 1.4

(Exponential Ergodicity) Let $\mathfrak {m}\in \mathbb {R}$ and $\delta \in (0,1/2)$. Then there exists a unique measure $\nu \in \mathcal {P}\big (\mathcal {C}^{1/2-\delta }_\mathfrak {m}(\mathbb {T})\big )$ which is invariant for the semi-group associated to (1.1). Furthermore, $\nu$ has full support in $\mathcal {C}^{1/2-\delta }_\mathfrak {m}$ and there exists $c>0$ such that for all $t>1$ and $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}$,

$$\begin{aligned} \Vert \mathcal {L}(u_t(\zeta )) -\nu \Vert _{\mathrm {TV}} \,\le e^{-ct}. \end{aligned}$$

(1.4)

Finally, for any $p\in [1,\infty )$ there exists $\Lambda := \Lambda (p,\delta )>0$ such that

$$\begin{aligned} \int _{\mathcal {C}^{1/2-\delta }_{\mathfrak {m}}} \exp \left( \Lambda \Vert f\Vert ^{1-2\delta }_{L^p}\right) \,\nu (\mathop {}\!\mathrm {d}f)<\infty . \end{aligned}$$

(1.5)

The proof of Theorem 1.1 is completed at the end of Sect. 4 and the proof of Theorem 1.4 is completed at the end of Sect. 5.

2 Stochastic heat equation

In what follows we will decompose the solution u to (1.1) into a regular and irregular part, with the irregular part being the solution to a linear stochastic heat equation (SHE). We briefly recap some necessary definitions of space-time white noise, the solution to the SHE and some of its properties.

Definition 2.1

(White Noise) Given an abstract probability space $(\Omega ,\mathcal {F},\mathbb {P})$ we say that an $\mathbb {R}$-valued, stochastic process, indexed by $L^2(\mathbb {R}_+\times \mathbb {T})$, $\{\xi (\varphi )\,:\,\varphi \in L^2(\mathbb {R}_+\times \mathbb {T})\}$, defines a spatially mean-free space-time white noise if

• 1. $\mathbb {E}[\xi (\varphi )]=0$, $\mathbb {E}[\xi (\varphi )\xi (\varphi ')] = \langle \varphi ,\varphi '\rangle _{L^2(\mathbb {R}_+\times \mathbb {T})}$ for all $\varphi ,\,\varphi ' \in L^2(\mathbb {R}_+\times \mathbb {T})$,

• 2. $\xi (\psi \otimes 1)=0$ for all $\psi \in L^2(\mathbb {R}_+)$, $\mathbb {P}$-a.s.

For $\varphi \in L^2(\mathbb {R}_+ \times \mathbb {T})$ we write $\xi (\varphi )$ in the convenient form of a stochastic integral,

$$\begin{aligned} \xi (\varphi ) = \int _{\mathbb {R}_+} \int _{\mathbb {T}} \varphi (t,x) \xi (\mathop {}\!\mathrm {d}t,\,\mathop {}\!\mathrm {d}x), \end{aligned}$$

(2.1)

even though $\xi$ is almost surely not a measure. We refer to [6, Ch. 4] for further details and a proper construction of (2.1).

We define the filtration

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

and let $(\mathcal {F}_t)_{t\ge 0}$ denote its usual augmentation.

Definition 2.2

Let $0\le t_0<T$ and $\mathcal {H}$ be the periodic heat kernel on $\mathbb {T}$ defined in (A.9). We say that the $\mathcal {S}'(\mathbb {T})$ valued process,

$$\begin{aligned} (t_0,T] \ni t\mapsto v_{t_0,t}(\phi ):=\int _{t_0}^t \int _{\mathbb {T}}(\mathcal {H}_{t-r}*\phi )(x)\,\xi (\mathop {}\!\mathrm {d}r,\mathop {}\!\mathrm {d}x) \quad \,\forall \phi \in C^\infty (\mathbb {T}), \end{aligned}$$

(2.2)

is a mild solution to the SHE, with zero initial condition at $t=t_0$.

Theorem 2.3

The process $[t_0,T]\ni t\mapsto v_{t_0,t}$ is, continuous, Markov, $(\mathcal {F}_t)_{t \in [t_0,T]}$ adapted and for any $\alpha <1/2$, $\kappa \in [0,1/2)$ and $p\ge 1$ there exists $C:=C(T,\alpha ,\kappa ,p)>0$ such that,

$$\begin{aligned} \mathbb {E}\left[ \sup _{t\,\ne \,s \in [t_0,T]} \frac{\Vert v_{t_0,t}-v_{t_0,s}\Vert ^p_{\mathcal {C}^{\alpha -2\kappa }}}{|t-s|^{p\kappa }} \right] <C. \end{aligned}$$

(2.3)

Furthermore, for any $0\le t_0 < t\le T$, the law of $v_{t_0,t}$ depends only on $|t-t_0|$.

Proof

See “Appendix B”. $\square$

3 Local well-posedness

Throughout this section, we fix $T>0$ and $\chi \in \mathbb {R}$ (not necessarily positive). From Theorem 2.3 we see that $\mathbb {P}$-a.s. the map $t\mapsto v_{0,t}$ is finite only in $C_T\mathcal {C}_0^{\alpha }$, for $\alpha <1/2$. We therefore cannot expect to find solutions to (1.1) of any higher regularity. Obtaining the a priori estimates in Sect. 4 requires at least one degree of spatial regularity. Therefore, although the equation is not singular itself, we solve for a remainder process with higher regularity using a Da Prato–Debussche trick, [4]. Concretely we decompose the solution as $u_t:=w_t+Z_t$, where $t\mapsto Z_t$ is a deterministic function taking values in the Hölder–Besov space $\mathcal {C}_0^{\alpha }$ with zero spatial mean. Subsequently we will take $Z_t$ to be a $\mathbb {P}$-a.s. realisation of the stochastic heat equation $t\mapsto v_{0,t}$. The unknown, w, solves,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t w- \partial _{xx} w = \chi \partial _x((w+Z)^2\partial _x \rho _{w+Z}), &{} \text{ on } \mathbb {R}_+ \times \mathbb {T},\\ -\partial _{xx} \rho _{w+Z} = w-\bar{w}+Z, &{}\text { on }\mathbb {T},\\ w|_{t=0}= \zeta , &{}\text { on }\mathbb {T}. \end{array}\right. } \end{aligned}$$

(3.1)

For the rest of this section, we fix $\zeta \in \mathcal {C}^{\alpha _0}(\mathbb {T})$ and $Z\in C_T\mathcal {C}^{\alpha }_0$.

We first show that under suitable regularity assumptions on w, the right hand side of the first equation in (3.1) is a well-defined element of $C_{\eta ;T}\mathcal {C}^\alpha (\mathbb {T})$.

Lemma 3.1

Let $w\in C_{\eta ;T}\mathcal {C}^{\alpha }$. Then the map $w\mapsto \Psi w$, defined for any $t\in (0,T]$, by

$$\begin{aligned} (\Psi w)_t := e^{t\Delta } \zeta + \int _0^t e^{(t-s)\Delta } \chi \partial _x ((w_s+Z_s)^2 \partial _x \rho _{w_s+Z_s})\mathop {}\!\mathrm {d}s, \end{aligned}$$

(3.2)

is well-defined from $C_{\eta ;T}\mathcal {C}^{\alpha }$ to itself.

Proof

Applying (A.10), for any $t >0$, we have that

$$\begin{aligned} \Vert e^{t\Delta }\zeta \Vert _{\mathcal {C}^{\alpha }} \lesssim t^{-\frac{\alpha -\alpha _0}{2}}\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}. \end{aligned}$$

(3.3)

Concerning the integral term, expanding the square and applying Theorem A.5, along with (A.12) we obtain the bounds

$$\begin{aligned} \begin{aligned} \Vert w^2_s\partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}&\le \Vert w\Vert ^2_{\mathcal {C}^{\alpha }}\Vert \partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}\lesssim \Vert w_s\Vert ^3_{\mathcal {C}^{\alpha }}+\Vert w_s\Vert ^2_{\mathcal {C}^{\alpha }}\Vert Z_s\Vert _{\mathcal {C}^{\alpha }}\\ 2\Vert w_sZ_s\partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}&\le 2\Vert w\Vert _{\mathcal {C}^{\alpha }}\Vert Z_s\Vert _{\mathcal {C}^{\alpha }}\Vert \partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}\\&\lesssim 2\Vert w_s\Vert ^2_{\mathcal {C}^\alpha }\Vert Z_s\Vert _{\mathcal {C}^\alpha } + 2\Vert w_s\Vert _{\mathcal {C}^{\alpha }}\Vert Z_s\Vert ^2_{\mathcal {C}^{\alpha }}\\ \Vert Z^2_s\partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}&\le \Vert Z_s\Vert ^2_{\mathcal {C}^{\alpha }}\Vert \partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}\lesssim \Vert w_s\Vert _{\mathcal {C}^{\alpha }}\Vert Z_s\Vert ^2_{\mathcal {C}^\alpha }+\Vert Z_s\Vert ^3_{\mathcal {C}^{\alpha }}. \end{aligned} \end{aligned}$$

Combining these yields that

$$\begin{aligned} \Vert (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}\lesssim \sum _{k=0}^{3}\left( {\begin{array}{c}3\\ k\end{array}}\right) \Vert w_s\Vert ^{3-k}_{\mathcal {C}^{\alpha }}\Vert Z_s\Vert ^k_{\mathcal {C}^{\alpha }}. \end{aligned}$$

(3.4)

Therefore, applying (A.10), (A.5) for any $s<t\in (0,T\wedge 1]$ we have

$$\begin{aligned}&\Vert e^{(t-s)\Delta }\partial _x ((w_s+Z_s)^2\partial _x \rho _{w_s+Z_s})\Vert _{\mathcal {C}^{\alpha }}\\&\quad \lesssim (t-s)^{-\frac{1}{2}}\Vert (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}\\&\quad \lesssim (t-s)^{-\frac{1}{2}}s^{-3\eta }\max _{k=0,\ldots ,3}\left\{ \Vert w\Vert ^{3-k}_{C_{\eta ;T}\mathcal {C}^{\alpha }}\Vert Z\Vert ^{k}_{C_T\mathcal {C}^{\alpha }} \right\} . \end{aligned}$$

Since $\eta <\frac{1}{4}$ we may integrate $s^{-3\eta }$ near 0 and so for $t \in (0,T\wedge 1]$ we have

$$\begin{aligned} t^\eta \Vert (\Psi w)_t\Vert _{\mathcal {C}^{\alpha }}\lesssim t^{\eta -\frac{\alpha -\alpha _0}{2}}\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}+t^{\frac{1}{2}-2\eta }\max _{k=0,\ldots ,3}\left\{ \Vert w\Vert ^{3-k}_{C_{\eta ;T}\mathcal {C}^{\alpha }}\Vert Z\Vert ^{k}_{C_T\mathcal {C}^{\alpha }} \right\} . \end{aligned}$$

(3.5)

where both exponents are positive due to (1.3) and the norms on the right hand side are finite by assumption. For $t>1$ one may argue in almost exactly the same way, only splitting the time integral at $t=1$ and replacing the multiplication by $t^\eta$ with $(t\wedge 1)^\eta$. $\square$

Definition 3.2

(Mild Solutions to (3.1)) We say that $w\in C_{\eta ;T}\mathcal {C}^\alpha$ is a mild solution to (3.1) on [0, T] (started from $\zeta$ and driven by Z) if for every $t\in (0,T]$,

$$\begin{aligned} w_t = e^{t\Delta }\zeta +\int _0^t e^{(t-s)\Delta }\chi \partial _x \left( (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s} \right) \,\mathop {}\!\mathrm {d}s. \end{aligned}$$

(3.6)

Remark 3.3

Lemma 3.1 demonstrates that for any solution, the right hand side of (3.6) is well-defined.

Theorem 3.4

(Local Well-Posedness of (3.1)) Let $\mathfrak {R}\ge 1$ be such that $\Vert Z\Vert _{C_T\mathcal {C}^{\alpha }}^3 + \Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}(\mathbb {T})}<\mathfrak {R}$. Then there exists $C>0$, independent of $\mathfrak {R}$, $\zeta$, and Z, such that (3.1) has a unique mild solution $w \in C_{\eta ;T_*}\mathcal {C}^\alpha$ where

$$\begin{aligned} T_* = \left( \frac{1}{C\mathfrak {R}}\right) ^{\frac{1}{\theta }}\wedge T, \,\, \text { with }\,\,\theta :=\left( \eta -\frac{\alpha -\alpha _0}{2}\right) \wedge \left( \frac{1}{2}-2\eta \right) . \end{aligned}$$

(3.7)

Furthermore

$$\begin{aligned} \sup _{t \in (0,T_*]} t^\eta \Vert w_t\Vert _{\mathcal {C}^{\alpha }} \le 1 \end{aligned}$$

(3.8)

and

$$\begin{aligned} \lim _{t\rightarrow 0} \Vert w_t-\zeta \Vert _{\mathcal {C}^{\alpha _0}}=0. \end{aligned}$$

(3.9)

Proof

Denoting

$$\begin{aligned} B_{T_*} := \Big \{ w \in C((0,T_*];\mathcal {C}^{\alpha }(\mathbb {T}))\,:\, \sup _{t \in (0,T_*]}t^\eta \Vert w_t\Vert _{\mathcal {C}^{\alpha }}\le 1\Big \}, \end{aligned}$$

we will show that

$$\begin{aligned} w \mapsto (\Psi w)_t:=e^{t\Delta }\zeta +\int _0^t e^{(t-s)\Delta }\chi \partial _x \left( (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s} \right) \,\mathop {}\!\mathrm {d}s, \end{aligned}$$

is a contraction on $B_{T_*}$ for $T_*$ defined by (3.7) for $C>0$ sufficiently large. By (3.5), for $w\in B_{T_*}$ and $t\in (0,T_*]$, there exists $C>0$ such that

$$\begin{aligned} t^\eta \Vert (\Psi w)_t\Vert _{\mathcal {C}^{\alpha }}&\le C t^{\left( \eta -\frac{\alpha -\alpha _0}{2}\right) \wedge \left( \frac{1}{2}-2\eta \right) }\mathfrak {R}, \end{aligned}$$

and so $\Psi$ maps $B_{T_*}$ into itself for $T_*$ defined by (3.7). To show that $\Psi$ is a contraction we let $w,\,\tilde{w} \in B_{T_*}$. For any $s\in (0,T_*]$, using similar steps to those in the proof of (3.5), we have that

$$\begin{aligned}&\Vert (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s}- (\tilde{w}_s+Z_s)^2\partial _x \rho _{\tilde{w}_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}\\&\quad \lesssim \Vert w_s-\tilde{w}_s\Vert _{\mathcal {C}^{\alpha }}\sum _{k=0}^2 \left( {\begin{array}{c}2\\ k\end{array}}\right) \left( \Vert w_s\Vert _{\mathcal {C}^{\alpha }}\vee \Vert \tilde{w}_s\Vert _{\mathcal {C}^{\alpha }}\right) ^{2-k}\Vert Z_s\Vert ^{k}_{\mathcal {C}^{\alpha }}\\&\quad \lesssim \mathfrak {R}s^{-2\eta }\Vert w_s-\tilde{w}_s\Vert _{\mathcal {C}^{\alpha }}. \end{aligned}$$

So then, for any $t \in (0,T_*]$, we have

$$\begin{aligned} t^\eta \Vert (\Psi w)_t-(\Psi \tilde{w})_t\Vert _{\mathcal {C}^{\alpha }}&\le t^\eta \int _0^t \Vert e^{(t-s)\Delta } \chi \partial _x \big ( (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s}\\&\quad -(\tilde{w}_s+Z_s)^2\partial _x \rho _{\tilde{w}_s+Z_s} \big ) \Vert _{\mathcal {C}^{\alpha }}\,\mathop {}\!\mathrm {d}s\\&\lesssim t^\eta \int _0^t (t-s)^{-\frac{1}{2}}\Vert (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s}\\&\quad -(\tilde{w}_s+Z_s)^2\partial _x \rho _{\tilde{w}_s+Z_s}\Vert _{\mathcal {C}^{\alpha }}\,\mathop {}\!\mathrm {d}s\\&\lesssim \mathfrak {R}t^{\frac{1}{2}-2\eta }\Vert w-\tilde{w}\Vert _{C_{\eta ;t}\mathcal {C}^{\alpha }}. \end{aligned}$$

It follows that there exists $C>0$ such that, for $T_*$ given by (3.7),

$$\begin{aligned} \Vert \Psi w -\Psi \tilde{w}\Vert _{C_{\eta ;T_*}\mathcal {C}^\alpha }&\le C\mathfrak {R}(T_*)^{\frac{1}{2}-2\eta }\Vert w - \tilde{w}\Vert _{C_{\eta ;T_*}\mathcal {C}^\alpha }\\&\le \frac{1}{2} \Vert w - \tilde{w}\Vert _{C_{\eta ;T_*}\mathcal {C}^\alpha }. \end{aligned}$$

Hence $\Psi$ is a contraction on $B_{T_*}$, and therefore there exists a unique fixed point $w \in B_{T_*}$ of $\Psi$ which, by construction, is a mild solution to (3.1) in the sense of Definition 3.2 and satisfies (3.8).

To show that w is the unique solution in all of $C_{\eta ;T_*}\mathcal {C}^{\alpha }$, let $\tilde{w}$ be another mild solutions of (3.1). Then, by a similar argument to the above, there exists a $\tilde{T}(\Vert w\Vert _{C_{\eta ;T_*}\mathcal {C}^\alpha },\Vert \tilde{w}\Vert _{C_{\eta ;T_*}\mathcal {C}^\alpha })=:\tilde{T}\in (0,T_*]$ such that $w,\tilde{w}\in B_{\tilde{T}}$. Since both must be fixed points of $\Psi$ on $B_{\tilde{T}}$ we have that $w=\tilde{w}$ on $[0,\tilde{T}]$. Iterating the argument, using the same $\tilde{T}$ at each step, shows that $w=\tilde{w}$ on $[0,T_*]$.

To show (3.9), observe first that $\lim _{t\rightarrow 0}\Vert e^{t\Delta }\zeta -\zeta \Vert _{\mathcal {C}^{\alpha _0}}=0$ by Remark A.7 and so it only remains to show that the integral term converges to zero in $\mathcal {C}^{\alpha _0}$. Consider $\tilde{\eta } \in \left( \frac{\alpha -\alpha _0}{2},\eta \right)$. Since $C_{\tilde{\eta };T_*}\mathcal {C}^{\alpha } \hookrightarrow C_{\eta ;T_*}\mathcal {C}^{\alpha }$, applying what we have proved so far to $(\alpha _0,\alpha ,\tilde{\eta })$ in place of $(\alpha _0,\alpha ,\eta )$, we see that, for $S>0$ sufficiently small, w is also the unique mild solution to (3.1) in $C_{\tilde{\eta };S}\mathcal {C}^\alpha$, and that $\sup _{t\in (0,S]}t^{\tilde{\eta }}\Vert w_t\Vert _{\mathcal {C}^\alpha } \le 1$. Applying (A.10) and (3.4), for all $t\in (0,S]$

$$\begin{aligned} \int _0^t \Vert e^{(t-s)\Delta } \partial _x((w_s+Z_s)^2\partial _x\rho _{w_s+Z_s}\Vert _{\mathcal {C}^{\alpha _0}}\mathop {}\!\mathrm {d}s&\lesssim \int _0^t (t-s)^{-\frac{\alpha _0 -\alpha +1}{2}} s^{-3\tilde{\eta }} \mathop {}\!\mathrm {d}s\\&= t^{\frac{1}{2}+\frac{\alpha -\alpha _0}{2}-3\tilde{\eta }}, \end{aligned}$$

where we used the fact that $\frac{\alpha _0-\alpha +1}{2}\vee 3\tilde{\eta }<1$ to evaluate the integral. We now choose $\tilde{\eta }$ sufficiently close to $\frac{\alpha -\alpha _0}{2}$ so that $\frac{1}{2}+\frac{\alpha -\alpha _0}{2}-3\tilde{\eta }>0$, from which (3.9) follows. $\square$

Lemma 3.5

Suppose that $w\in C_{\eta ;T}\mathcal {C}^\alpha$ is a mild solution to (3.1). Then for all $t_0\in (0,T)$, $\beta \in (\alpha ,\alpha +1)$ and $\kappa \in (0,1)$,

$$\begin{aligned} \sup _{t\,\ne \, s\, \in [t_0,T]}\frac{\Vert w_t-w_s \Vert _{\mathcal {C}^{\beta -2\kappa }}}{|t-s|^{\kappa }}<\infty . \end{aligned}$$

(3.10)

Proof

Applying (A.10) gives

$$\begin{aligned} \Vert e^{t\Delta }\zeta \Vert _{\mathcal {C}^\beta }\lesssim t^{-\frac{\beta -\alpha _0}{2}}\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}. \end{aligned}$$

(3.11)

Regarding the integral term, applying similar steps to those in the proof of (3.5) and the assumption $\frac{1+\beta -\alpha }{2}<1$,

$$\begin{aligned} \int _{0}^t\Vert e^{(t-s)\Delta }\partial _x \left( (w_s+Z_{s})^2\partial _x \rho _{w_s+Z_s}\right) \Vert _{\mathcal {C}^{\beta }}\,\mathop {}\!\mathrm {d}s&\lesssim \int _{0}^t (t-s)^{-\frac{1+\beta -\alpha }{2}}s^{-3\eta }\,\mathop {}\!\mathrm {d}s \nonumber \\&= t^{\frac{1}{2}-\frac{\beta -\alpha }{2}-3\eta } . \end{aligned}$$

(3.12)

It follows that

$$\begin{aligned} \sup _{t \in [t_0,T_*]} \Vert w_t\Vert _{\mathcal {C}^{\beta }} <\infty . \end{aligned}$$

Now, let $0<t_0<s<t\le T$ and $\kappa \in [0,1)$. Applying the triangle inequality and using the semi-group property, we have that

$$\begin{aligned} \begin{aligned} \Vert w_t-w_s\Vert _{\mathcal {C}^{\beta -2\kappa }}&\le \left\| (e^{(t-s)-1)\Delta }w_{s}\right\| _{\mathcal {C}^{\beta -2\kappa }} \\&\quad + \int _{s}^{t} \left\| e^{(t-r)\Delta }\partial _x ((w_{r}+Z_{r})^2\partial _x \rho _{w_{r}+Z_{r}})\right\| _{\mathcal {C}^{\beta -2\kappa }} \,\mathop {}\!\mathrm {d}r\\&=: I_1(s,t)+\int _s^{t} I_2(r,t)\,\mathop {}\!\mathrm {d}r. \end{aligned} \end{aligned}$$

Using (A.10) and (A.11),

$$\begin{aligned} I_1(s,t) \lesssim \left\| \left( e^{(t-s)\Delta } - 1\right) w_{s}\right\| _{\mathcal {C}^{\beta -2\kappa }} \lesssim (t-s)^{\kappa }\Vert w_{s}\Vert _{\mathcal {C}^\beta }. \end{aligned}$$

(3.13)

Since $\beta -1<\alpha$ and $\sup _{t \in [t_0,T_*]}\Vert Z_t\Vert _{\mathcal {C}^\alpha }+\Vert w_t\Vert _{\mathcal {C}^{\alpha }}<\infty$, applying (A.10) followed by (A.11) and (3.4), we obtain

$$\begin{aligned} I_2(r,t) \lesssim (t-r)^{-\frac{1+\beta -2\kappa -(\beta -1)}{2}}\left\| ((w_{r}+Z_{r})^2\partial _x \rho _{w_{r}+Z_{r}})\right\| _{\mathcal {C}^{\beta -1}}\lesssim (t-r)^{-1+\kappa }. \end{aligned}$$

Therefore

$$\begin{aligned} \int _s^{t} I_2(r,t)\,\mathop {}\!\mathrm {d}r \lesssim (t-s)^{\kappa }. \end{aligned}$$

(3.14)

Combining (3.13) and (3.14) we obtain (3.10). $\square$

Lemma 3.6

(Mild Solutions are Weak Solutions) Let $t_0 \in [0,T)$ and $w\in C_{\eta ;T}\mathcal {C}^{\alpha }$ be a mild solution to (3.1) on [0, T]. Then for any $\phi \in C^\infty (\mathbb {T})$ and $t \in [t_0,T]$,

$$\begin{aligned} \langle w_t,\phi \rangle - \langle w_{t_0},\phi \rangle = -\int _{t_0}^t \langle \partial _x w_s +\chi (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s},\partial _x \phi \rangle \,\mathop {}\!\mathrm {d}s. \end{aligned}$$

(3.15)

Moreover, (3.15) holds for all $t_0\in (0,T)$ and $\phi \in C^1(\mathbb {T})$.

Proof

From the mild form of the equation started from data $w_{t_0}$ at $t=t_0$, it follows that for $\phi \in C^{\infty }(\mathbb {T})$

$$\begin{aligned} \int _{t_0}^t \langle w_s,\partial _{xx} \phi \rangle \,\mathop {}\!\mathrm {d}s&= \int _{t_0}^t \left\langle e^{s\Delta }w_{t_0},\partial _{xx} \phi \right\rangle \,\mathop {}\!\mathrm {d}s\\&\quad + \int _{t_0}^t \int _{t_0}^s\left\langle e^{(s-r)\Delta } \chi \partial _x ((w_r+Z_r)^2\partial _x \rho _{w_r+Z_r}),\partial _{xx} \phi \right\rangle \,\mathop {}\!\mathrm {d}r\,\mathop {}\!\mathrm {d}s. \end{aligned}$$

Integrating by parts in the first term on the right hand side gives

$$\begin{aligned} \int _{t_0}^t \left\langle e^{s\Delta }w_{t_0},\partial _{xx} \phi \right\rangle \,\mathop {}\!\mathrm {d}s = \int _{t_0}^t \langle \partial _{xx} e^{s\Delta }w_{t_0},\phi \rangle \,\mathop {}\!\mathrm {d}s = \langle e^{t\Delta }w_{t_0},\phi \rangle -\langle w_{t_0},\phi \rangle . \end{aligned}$$

(3.16)

Using the same argument for the second term on the right hand side we have that

$$\begin{aligned} \left\langle e^{(s-r)\Delta } \partial _x ((w_r+Z_r)^2\partial _x \rho _{w_r+Z_r}),\partial _{xx} \phi \right\rangle = \partial _s \left\langle e^{(s-r)\Delta } \partial _x ((w_r+Z_r)^2\partial _x \rho _{w_r+Z_r}), \phi \right\rangle . \end{aligned}$$

Changing the order of integration gives

$$\begin{aligned}&\int _{t_0}^t\int _{t_0}^s\langle e^{(s-r)\Delta } \partial _x ((w_r+Z_r)^2\partial _x \rho _{w_r+Z_r}),\partial _{xx} \phi \rangle \,\mathop {}\!\mathrm {d}r\,\mathop {}\!\mathrm {d}s\\&\quad =\int _{t_0}^t \langle e^{(t-r)\Delta } \partial _x ((w_t+Z_t)^2\partial _x \rho _{w_t+Z_t}),\phi \rangle \,\mathop {}\!\mathrm {d}r\\&\qquad - \int _{t_0}^t \langle \partial _x ((w_r+Z_r)^2\partial _x \rho _{w_r+Z_r}),\phi \rangle \,\mathop {}\!\mathrm {d}r. \end{aligned}$$

Putting this together with (3.16) gives that

$$\begin{aligned} \int _{t_0}^t \langle w_s,\partial _{xx} \phi \rangle \,\mathop {}\!\mathrm {d}s&= \langle e^{t\Delta }w_{t_0},\phi \rangle -\langle w_{t_0},\phi \rangle \\&\quad + \int _{t_0}^t \langle e^{(t-r)\Delta } \chi \partial _x ((w_r+Z_r)^2\partial _x \rho _{w_r+Z_r}),\phi \rangle \,\mathop {}\!\mathrm {d}r\\&\quad - \int _{t_0}^t \langle \chi \partial _x ((w_r+Z_r)^2\partial _x \rho _{w_r+Z_r}),\phi \rangle \,\mathop {}\!\mathrm {d}r. \end{aligned}$$

Rearranging and integrating the left hand side by parts once proves (3.15) for any $\phi \in C^\infty (\mathbb {T})$. For $t_0\in (0,T)$, by Theorem 3.4, we have that $\sup _{t\in [t_0,T]}\Vert w_t\Vert _{C^1}<\infty$, so that the map $C^1(\mathbb {T})\ni \phi \mapsto \int _{t_0}^t \langle \partial _x w_s + (w_s+Z_s)^2\partial _x\rho _{w_s+Z_s},\partial _x \phi \rangle \mathop {}\!\mathrm {d}s$ is continuous. Approximating $\phi \in C^1(\mathbb {T})$ by a smooth sequence gives the result. $\square$

It follows from the proof of Lemma 3.6 that any solution to (3.1) has constant spatial mean.

Corollary 3.7

Let $t\in [0,T]$ and $w \in C_{\eta ;T}\mathcal {C}^{\alpha }$ be a mild solution to (3.1). Then $\bar{w}_t = \bar{\zeta }$.

Proof

Applying Lemma 3.6 with $\phi \equiv 1$ and $t_0=0$ gives $\bar{w}_t=\langle w_t,1\rangle = \langle \zeta ,1\rangle =\bar{\zeta }$. $\square$

4 A priori estimate and global well-posedness

In this section we make use of the specific sign choice $\chi >0$ in (1.1) to obtain an a priori estimate on the solution in Theorem 4.4. Throughout this section we fix $T>0$, and a zero spatial mean function $Z \in C_T\mathcal {C}^{\alpha }_0$.

Proposition 4.1

(Testing the Equation) Let $0<t_0<t\le T$, $p\ge 2$ an integer, $\zeta \in \mathcal {C}^{\alpha _0}(\mathbb {T})$, and $w(\zeta ) \in C_{\eta ;T}\mathcal {C}^{\alpha }$ be a mild solution to (3.1) on [0, T]. Then

$$\begin{aligned} \begin{aligned} \frac{1}{p(p-1)}\left( \Vert w_t\Vert _{L^p}^p -\Vert w_{t_0}\Vert _{L^p}^p\right)&= -\int _{t_0}^t \langle w_s^{p-2}\partial _x w_s,\partial _x w_s\rangle \,\mathop {}\!\mathrm {d}s\\&\quad -\int _{t_0}^t\langle \chi (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s},w_s^{p-2}\partial _x w_s\rangle \,\mathop {}\!\mathrm {d}s. \end{aligned} \end{aligned}$$

(4.1)

Proof

Let $\beta \in (1,\alpha +1)$ and $\kappa \in \left( 1/2,\beta /2\right)$. By Lemma 3.5,

$$\begin{aligned} \sup _{s\,\ne \,r \in [t_0,T]} \frac{\Vert w_s-w_r\Vert _{L^\infty }}{|s-r|^\kappa }\le \sup _{s\,\ne \,r \in [t_0,T]} \frac{\Vert w_s-w_r\Vert _{\mathcal {C}^{\beta -2\kappa }}}{|s-r|^\kappa } < \infty . \end{aligned}$$

(4.2)

Now, let $\varphi \in C^{\infty }(\mathbb {R})$ and observe that for any $t_0\le r<s\le T$ we have the identity

$$\begin{aligned} \langle w_s,\varphi (w_s)\rangle - \langle w_r,\varphi (w_r)\rangle&= \langle w_s,\varphi (w_r)\rangle - \langle w_r,\varphi (w_r)\rangle + \langle w_s,\varphi (w_s)-\varphi (w_r)\rangle . \end{aligned}$$

So for any $t\in (t_0,T]$, and $n\ge 2$, defining a family of partitions, by setting $t_i = t_0+i(t-t_0)/n$, for $i=0,\ldots ,n$ and applying Lemma 3.6 with the test function $\varphi (w_{t_i})$, we have

$$\begin{aligned}&\langle w_t,\varphi (w_t)\rangle - \langle w_{t_0},\varphi (w_{t_0})\rangle \\&\quad =-\sum _{i=0}^{n-1}\int _{t_i}^{t_{i+1}} \langle \partial _x w_s +\chi (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s},\partial _x \varphi (w_{t_i})\rangle \mathop {}\!\mathrm {d}s \\&\qquad +\sum _{i=0}^{n-1} \left\langle w_{t_{i+1}},\varphi (w_{t_{i+1}})-\varphi (w_{t_i})\right\rangle \\&\quad =:-I_n(t)+R_n(t) . \end{aligned}$$

By continuity of $s\mapsto \partial _x\varphi (w_s)$ we see that

$$\begin{aligned} \lim _{n\rightarrow \infty } I_n(t)= \int _0^t \langle \partial _x w_s + \chi (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s},\partial _x \varphi (w_{s})\rangle \mathop {}\!\mathrm {d}s. \end{aligned}$$

(4.3)

For $R^n(t)$, Taylor’s formula gives

$$\begin{aligned} R_n(t)&= \sum _{i=0}^{n-1}\left[ \langle w_{t_{i+1}},(w_{t_{t+i}}-w_{t_i})\varphi '( w_{t_i})\rangle + \Big \langle w_{t_{i+1}},\int _{w_{t_i}}^{w_{t_{i+1}}} \varphi ''(y)(w_{t_{i+1}}-y)\,\mathop {}\!\mathrm {d}y\Big \rangle \right] \\&=:R_{n;1}(t)+R_{n;2}(t). \end{aligned}$$

We show that $R^n_2(t)$ converges to zero. Using (4.2), we have the bound

$$\begin{aligned} |R_{n;2}(t)|&\lesssim \sup _{s\in [t_0,t]}\Vert w_s\Vert _{L^\infty }\sup _{\begin{array}{c} i = 0,\ldots ,n-1\\ y \in [w_{t_{i}},w_{t_{i+1}}] \end{array}}|\varphi ''(y)|\,\sum _{i=0}^{n-1} \Vert w_{t_{i+1}}-w_{t_i}\Vert ^2_{L^\infty }\\&\overset{\tiny (4.2)}{\lesssim } \sum _{i=0}^{n-1} |t_{i+1}-t_i|^{2\kappa }, \end{aligned}$$

where we used that $\varphi ''$ is continuous. Regarding $R_{n;1}(t)$, we may apply Lemma 3.6 once again to each bracket, this time with the test function $w_{t_{i+1}}\varphi '(w_{t_i})$, to give that

$$\begin{aligned} R_{n;1}(t)&= -\sum _{i=0}^{n-1} \int _{t_i}^{t_{i+1}} \langle \partial _x w_s +\chi (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s},\partial _x (w_{t_{i+1}}\varphi '(w_{t_i}))\rangle \,\mathop {}\!\mathrm {d}s. \end{aligned}$$

So again, by regularity of the map $s \mapsto \partial _x w_s$ and $\varphi$, we have that

$$\begin{aligned} \lim _{n\rightarrow \infty }R_n(t)= -\int _{t_0}^{t} \langle \partial _x w_s +\chi (w_s+Z_s)^2\partial _x \rho _{w_s+Z_s},\partial _x (w_{s}\varphi '(w_{s}))\rangle \,\mathop {}\!\mathrm {d}s. \end{aligned}$$

(4.4)

So combining (4.3) and (4.4) gives,

$$\begin{aligned} \langle w_t,\varphi (w_t)\rangle - \langle w_{t_0},\varphi (w_{t_0})\rangle&= -\int _{t_0}^t \langle \partial _x w_r + \chi (w_r+Z_r)^2\partial _x \rho _{w_r+Z_r},\\&\quad \times \left( 2\varphi '(w_r)+ w_r \varphi ''(w_r)\right) \partial _x w_r \rangle \,\mathop {}\!\mathrm {d}r. \end{aligned}$$

For $\varphi (x):= x^{p-1}$ we have $2\varphi '(x) + x\varphi ''(x)= p(p-1)x^{p-2}$ and

$$\begin{aligned} \langle w_t,w_t^{p-1}\rangle - \langle w_{t_0},w_{t_0}^{p-1}\rangle = \Vert w_t\Vert _{L^p}^p-\Vert w_{t_0}\Vert _{L^p}^p, \end{aligned}$$

which gives (4.1). $\square$

Observe that one of the terms appearing in the second term on the right hand side of (4.1) is

$$\begin{aligned} -\langle \partial _x \rho _{w},w^p\partial _x w\rangle = -\frac{1}{p+1}\langle \partial _x \rho _{w},\partial _x w^{p+1}\rangle = -\frac{1}{p+1}\Vert w^{p+2}\Vert _{L^1} + \frac{1}{p+2}\langle \bar{w},w^{p+1} \rangle . \end{aligned}$$

We will use this term in combination with the first term, $-\Vert w_t^{p-2}|\partial _x w_t|^2\Vert _{L^1}$, coming from the Laplacian, to obtain an a priori estimate in Theorem 4.4 on $\Vert w_{t}\Vert ^p_{L^p}$ that is independent of the initial data. To do so we make use of an ODE comparison lemma which can be found with proof as [34, Lem. 3.8].

Lemma 4.2

(ODE Comparison) Let $\lambda >1$, $c_1,\,c_2>0$ and $f:[0,T]\rightarrow [0,\infty )$ be differentiable and satisfying

$$\begin{aligned} f'(t) + c_1 f(t)^\lambda \le c_2, \end{aligned}$$

for every $t\in [0,T]$. Then for all $t\in (0,T]$

$$\begin{aligned} f(t)\le \max \Big \{ \Big (\frac{tc_1(\lambda -1)}{2}\Big )^{-\frac{1}{\lambda -1}},\,\left( \frac{c_2}{c_1}\right) ^{\frac{1}{\lambda }}\Big \} . \end{aligned}$$

Remark 4.3

For a given initial data $\zeta \in \mathcal {C}^{\alpha _0}(\mathbb {T})$, a mild solution w on [0, T] to (3.1), and a constant $c\in \mathbb {R}$ with $c\ne 0$, we have that $\tilde{w}_t=cw_t$ satisfies

$$\begin{aligned} \tilde{w}_t = e^{t\Delta }\tilde{\zeta }+ \int _0^t e^{(t-s)\Delta } c^{-2}\chi \partial _x \big ( (\tilde{w}_s + \tilde{Z}_s)^2\partial _x\rho _{\tilde{w}_s+\tilde{Z}_s} \big )\mathop {}\!\mathrm {d}s\;, \end{aligned}$$

where $\tilde{\zeta }= c\zeta$ and $\tilde{Z}_s = cZ_s$. In particular, if $\chi >0$, then setting $c:=\chi ^{1/2}$ we see that any solution to (3.1) is equal, up to a scalar multiple, to a solution to (3.1) with $(\zeta ,Z,\chi )$ replaced by $(\tilde{\zeta },\tilde{Z},1)$. Conversely, if $\bar{\zeta }\ne 0$, then setting $c:=1/\bar{\zeta }$ we see that any solution to (3.1) is equal, up to a scalar multiple, to a solution to (3.1) with $(\zeta ,Z,\chi )$ replaced by $(\tilde{\zeta },Z,c^2 \chi )$ and where now $\tilde{\zeta }\in \mathcal {C}^{\alpha _0}_1(\mathbb {T})$.

In light of Remark 4.3, we phrase the following a priori estimate only for the case $\bar{\zeta }\in \{0,1\}$.

Theorem 4.4

(A Priori Bound on the Remainder) Let $p\ge 2$ be an even integer, $\chi >0$, $\mathfrak {m}\in \{0,1\}$, $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$ and $w(\zeta ) \in C_{\eta ;T}\mathcal {C}^{\alpha }$ be a mild solution to (3.1) on [0, T]. Then there exists a constant $C>0$, independent of $\zeta$, Z, p and $\chi$, such that

$$\begin{aligned} \Vert w_t(\zeta )\Vert _{L^p} \le \max \Big \{ (\chi t/4)^{-\frac{1}{2}}, C(p\vee \chi )^{\gamma }\Vert Z\Vert ^{\frac{1}{\alpha }}_{C_t\mathcal {C}^{\alpha }},\,C(p\vee \chi )^{\frac{p+1}{p+2}}\mathfrak {m}\, \Big \}, \end{aligned}$$

(4.5)

where $\gamma =\frac{p(1+\alpha )+2+\alpha }{\alpha (p+2)}$.

Proof

Since $t\mapsto \partial _x w_t$ and $t\mapsto (w_t+Z_t)^2\partial _x \rho _{w_t+Z_t}$ are both continuous mappings from (0, T] into $L^\infty (\mathbb {T})$ by Lemma 3.5, we may differentiate (4.1) with respect to t in order to obtain

$$\begin{aligned} \frac{1}{p(p-1)}\frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}t} \Vert w_t\Vert _{L^p}^p = - \Vert w_t^{p-2}|\partial _x w_t|^2\Vert _{L^1}-\chi \langle (w_t+Z_t)^2\partial _x \rho _{w_t+Z_t},w_t^{p-2}\partial _x w_t\rangle , \end{aligned}$$

where we used that p is even in the first term on the right hand side. Regarding the second term, since $f\mapsto \partial _x \rho _f$ is linear, we have

$$\begin{aligned} \langle (w_t+Z_t)^2\partial _x \rho _{w_t+Z_t},w_t^{p-2}\partial _x w_t\rangle&= \langle \partial _x \rho _{w_t}, w^p_t\partial _x w_t\rangle + \langle \partial _x\rho _{Z_t}, w^p_t \partial _x w_t\rangle \\&\quad + 2\langle Z_t\partial _x \rho _{w_t+Z_t},w^{p-1}_t\partial _x w_t\rangle \\&\quad + \langle Z_t^2\partial _x\rho _{w_t+Z_t},w^{p-2}_t\partial _x w_t\rangle . \end{aligned}$$

Integrating the first term by parts, using that $-\partial _{xx}\rho _{w_t}=w_t-\bar{\zeta }$ by Corollary 3.7 and recalling that we have set $\bar{\zeta } = \mathfrak {m}$,

$$\begin{aligned} \langle \partial _x \rho _{w_t}, w^p_t\partial _x w_t\rangle&=\frac{1}{p+1}\langle \partial _x \rho _{w_t}, \partial _x w^{p+1}_t\rangle = \frac{1}{p+1}\langle w_t-\mathfrak {m},w_t^{p+1}\rangle \\&= \frac{1}{p+1} \Vert w_t^{p+2}\Vert _{L^1} -\frac{1}{p+1}\langle \mathfrak {m},w_t^{p+1}\rangle , \end{aligned}$$

Therefore, applying the chain rule in the remaining terms gives

$$\begin{aligned} \frac{1}{p(p-1)\chi }\frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}t} \Vert w_t\Vert _{L^p}^p&= - \frac{1}{\chi }\Vert w_t^{p-2}|\partial _x w_t|^2\Vert _{L^1} - \frac{1}{p+1} \Vert w_t^{p+2}\Vert _{L^1}+ \frac{1}{p+1}\langle \mathfrak {m},w_t^{p+1}\rangle \nonumber \\&\quad - \frac{1}{p+1}\langle \partial _x\rho _{Z_t}, \partial _x w^{p+1}_t\rangle - \frac{2}{p}\langle Z_t\partial _x \rho _{Z_t},\partial _x w^p_t\rangle \nonumber \\&\quad - \frac{1}{p-1}\langle Z_t^2\partial _x\rho _{Z_t},\partial _x w^{p-1}_t\rangle - \frac{2}{p}\langle Z_t,\partial _x \rho _{w_t}\partial _x w^p_t\rangle \nonumber \\&\quad -\frac{1}{p-1}\langle Z_t^2,\partial _x\rho _{w_t}\partial _x w^{p-1}_t\rangle . \end{aligned}$$

(4.6)

We demonstrate that all terms without a definite sign can be bounded by a constant multiple of the two negative terms,

$$\begin{aligned} \frac{1}{\chi }\Vert w_t^{p-2}|\partial _xw_t|^2\Vert _{L^1}+\frac{1}{p+1}\Vert w_t^{p+2}\Vert _{L^1}=: \frac{1}{\chi }A_t + \frac{1}{p+1}B_t. \end{aligned}$$

From Theorems A.2 and A.8, for any $\alpha \in \mathbb {R}$ and $p,q \in [1,\infty ]$, we have that

$$\begin{aligned} \Vert \partial _x \rho _w\Vert _{\mathcal {B}^{\alpha }_{p,q}} \lesssim \Vert w\Vert _{\mathcal {B}^{\alpha -\frac{1}{p}}_{1,q}},\quad \text { and }\quad \Vert \partial _x \rho _w\Vert _{L^\infty } \lesssim \Vert w\Vert _{L^1}. \end{aligned}$$

(4.7)

For any $0<q\le p+2$, by Jensen’s inequality we have

$$\begin{aligned} \Vert w_t^q\Vert _{L^1}\le B_t^{\frac{q}{p+2}}. \end{aligned}$$

(4.8)

Furthermore, by applying Cauchy–Schwarz followed by (4.8), for $\frac{2}{p}< q \le p+1$ we have

$$\begin{aligned} \Vert \partial _x w^q_t\Vert _{L^1} = q\Vert w^{q-1}_t \partial _x w_t\Vert _{L^1} \le q\Vert w^{2q-p}_t\Vert ^{\frac{1}{2}}_{L^1}\,A_t^{\frac{1}{2}}\le qB_t^{\frac{2q-p}{2(p+2)}}\,A_t^{\frac{1}{2}}. \end{aligned}$$

(4.9)

To keep track of dependence on p, we also make use of the following inequality which readily follows from Young’s inequality for products: for $\gamma _1,\gamma _2 \in (1,\infty )$ such that $\frac{1}{\gamma _1}+\frac{1}{\gamma _2}=1$, $a,\,b>0$ and $c\ge 1$

$$\begin{aligned} ab < \frac{c^{\gamma _1-1}}{\gamma _1-1} a^{\gamma _1} + \frac{1}{c}b^{\gamma _2} \end{aligned}$$

(4.10)

(in fact $ab < \frac{c^{\gamma _1-1}}{e(\gamma _1-1)} a^{\gamma _1} + \frac{1}{c}b^{\gamma _2}$). From now on we let $C>0$, $c\ge 1$ be constants, that are independent of $\zeta ,\,Z,\,p,$ and $\chi$. Later, we will fix $c\ge 1$ sufficiently large at the end of the proof. If we write $\lesssim$ in an inequality below, the implied proportionality constant is equal to $C>0$ which we take sufficiently large so that the inequality holds.

We work through the terms of (4.6) in order. For the third term of (4.6), applying Hölder, (4.8), (4.10) and using that $\mathfrak {m}\in \{0,1\}$, we have,

$$\begin{aligned} \frac{1}{p+1}|\langle \mathfrak {m}, w^{p+1}_t\rangle | \le \frac{\mathfrak {m}}{p+1} \Vert w^{p+1}_t\Vert _{L^1} \le \frac{\mathfrak {m}}{p+1}B_t^{\frac{p+1}{p+2}} \le \frac{\mathfrak {m}c^{p+1}}{(p+1)^2} + \frac{1}{c(p+1)}B_t. \end{aligned}$$

(4.11)

For the fourth term of (4.6), we integrate by parts to give

$$\begin{aligned} \begin{aligned} \frac{1}{p+1}|\langle \partial _x \rho _{Z_t}, \partial _x w_t^{p+1}\rangle | =\frac{1}{p+1} |\langle Z_t,w_t^{p+1}\rangle |\;&\overset{(\mathrm{A.2})\, (4.8)}{\lesssim }\frac{1}{p} \Vert Z\Vert _{C_t\mathcal {C}^{\alpha }} B_t^{\frac{p+1}{p+2}}\\&\quad \overset{(4.10)}{\le }\; \frac{c^{p+1}}{p^2} \Vert Z\Vert _{C_t\mathcal {C}^{\alpha }}^{p+2} + \frac{1}{cp}B_t. \end{aligned} \end{aligned}$$

(4.12)

Concerning the two remaining terms of (4.6) involving $\partial _x \rho _{Z_t}$, we have for $k=1,2$

$$\begin{aligned} \begin{aligned} \frac{1}{p+1-k}|\langle Z^k_t\partial _x \rho _{Z_t},\partial _x w^{p+1-k}_{t} \rangle |&\le \frac{1}{p+1-k}\Vert Z^k_t\partial _x \rho _{Z_t}\Vert _{L^\infty }\, \Vert \partial _x w^{p+1-k}_{t} \Vert _{L^1}\\&\overset{\mathrm{(A.7)},\,\mathrm{(A.13)},\,(4.9)}{\lesssim } \Vert Z_t\Vert ^{k+1}_{\mathcal {C}^\alpha }B_t^{\frac{p+2-2k}{2(p+2)}}A_t^{\frac{1}{2}}\\&\overset{(4.10)}{ \le } c \Vert Z_t\Vert ^{2(k+1)}_{\mathcal {C}^{\alpha }}\,B_t^{\frac{p+2-2k}{p+2}} + \frac{1}{c} A_t\\&\overset{(4.10)}{ \lesssim } \frac{c^{\frac{p+2-k}{k}}}{p} \Vert Z\Vert ^{\frac{k+1}{k}(p+2) }_{C_t\mathcal {C}^{\alpha }} + \frac{1}{c}B_t + \frac{1}{c} A_t. \end{aligned} \end{aligned}$$

(4.13)

(In the case $p=k=2$, note that we do not need to apply (4.10) in the final line.) Combining (4.12), (4.13) and noting that $\frac{3}{2}(p+2)$ is the highest power of $\Vert Z\Vert _{C_t\mathcal {C}^{\alpha }}$ encountered so far, we have that

$$\begin{aligned} \begin{aligned} \sum _{k=0}^2 \left( {\begin{array}{c}2\\ k\end{array}}\right) \frac{1}{p+1-k} |\langle Z^k_t\partial _x \rho _{Z_t},\partial _x w^{p+1-k}_{t} \rangle |&\lesssim \frac{c^{p+1}}{p}\Vert Z\Vert ^{\frac{3}{2}(p+2)}_{C_t\mathcal {C}^{\alpha }} + \frac{1}{c}B_t +\frac{1}{c}A_t. \end{aligned} \end{aligned}$$

(4.14)

For the seventh term of (4.6), we first integrate by parts and apply the triangle inequality to obtain

$$\begin{aligned} \frac{1}{p}|\langle Z_t,\partial _x \rho _{w_t}\partial _x w_t^p\rangle |\, \le \, \frac{1}{p}|\langle \partial _x Z_t ,w^p_t\partial _x \rho _{w_t}\rangle | + \frac{1}{p}|\langle Z_t,w_t^{p+1}\rangle |+\frac{1}{p}|\langle Z_t,w^{p}_t\mathfrak {m}\rangle |. \end{aligned}$$

(4.15)

Since $\alpha \in (0,\frac{1}{2})$, using Theorems A.5 and A.4, we have

$$\begin{aligned} \Vert w_t^p\partial _x \rho _{w_t}\Vert _{\mathcal {B}^{1-\alpha }_{1,1}}&\overset{{(\mathrm A.8)} }{\lesssim } \Vert w_t^p\Vert _{L^{\frac{1}{1-\alpha }}}\Vert \partial _x \rho _{w_t}\Vert _{\mathcal {B}^{1-\alpha }_{\frac{1}{\alpha },1}} + \Vert w^p_t\Vert _{\mathcal {B}^{1-\alpha }_{1,1}}\Vert \partial _x \rho _{w_t}\Vert _{L^\infty }\nonumber \\&\overset{(4.7),\,{(\mathrm A.3)},\,{(\mathrm A.4)}}{\lesssim } \Vert w_t^p\Vert _{\mathcal {B}^{1-\alpha }_{1,1}}\Vert w_t\Vert _{L^1}\nonumber \\&\overset{{(\mathrm A.6)}}{\lesssim }\left( \Vert \partial _x w_t^p\Vert ^{1-\alpha }_{L^1}\Vert w_t^p\Vert ^\alpha _{L^1} + \Vert w^p_t\Vert _{L^1}\right) \Vert w_t\Vert _{L^1} \nonumber \\&\overset{(4.8)}{\lesssim } \Vert \partial _x w_t^p\Vert ^{1-\alpha }_{L^1}B_t^{\frac{p\alpha +1}{p+2}} + B_t^{\frac{p+1}{p+2}} \nonumber \\&\overset{(4.9)}{\lesssim } p^{1-\alpha }B_t^{\frac{p(1+\alpha )+2}{2(p+2)}}A_t^{\frac{1-\alpha }{2}} + B_t^{\frac{p+1}{p+2}} . \end{aligned}$$

(4.16)

Considering the first term of (4.15), applying (4.16) and then (4.10) twice with $\gamma _1=\frac{2}{1+\alpha }$ and then with $\gamma _1=\frac{(p+2)(1+\alpha )}{2\alpha }$, and using that $p^{-\alpha }\le 1$,

$$\begin{aligned} \frac{1}{p}|\langle \partial _x Z_t ,w^p_t\partial _x \rho _{w_t}\rangle |&\overset{(\mathrm{A.2})}{\lesssim } \frac{1}{p}\Vert \partial _x Z_t \Vert _{\mathcal {C}^{\alpha -1}}\Vert w_t^p\partial _x \rho _{w_t}\Vert _{\mathcal {B}^{1-\alpha }_{1,1}} \nonumber \\&\overset{(4.16)}{\lesssim } \Vert Z_t \Vert _{\mathcal {C}^{\alpha }} \left( B_t^{\frac{p(1+\alpha )+2}{2(p+2)}}A_t^{\frac{1-\alpha }{2}} + \frac{1}{p}B_t^{\frac{p+1}{p+2}} \right) \nonumber \\&\overset{(4.10) }{\lesssim }c \Vert Z\Vert ^{\frac{2}{1+\alpha }}_{C_t\mathcal {C}^{\alpha }}B_t^{\frac{p(1+\alpha )+2}{(p+2)(1+\alpha )}}+ \frac{c^{p+1}}{p^2}\Vert Z\Vert ^{p+2}_{C_t\mathcal {C}^{\alpha }} \nonumber \\&\qquad +\frac{1}{c}A_t + \frac{1}{pc}B_t\nonumber \\&\overset{(4.10)}{\lesssim } \frac{c^{\frac{p(1+\alpha )+2+\alpha }{\alpha }} }{p}\Vert Z\Vert ^{\frac{p+2}{\alpha }}_{C_t\mathcal {C}^{\alpha }} + \frac{p+1}{cp}B_t+\frac{1}{c}A_t. \end{aligned}$$

(4.17)

Concerning the second term of (4.15), we have

$$\begin{aligned} \begin{aligned} \frac{1}{p}|\langle Z_t,w_t^{p+1}\rangle |&\overset{{(\mathrm A.2)} ,\,(4.8)}{\lesssim } \frac{1}{p} \Vert Z\Vert _{C_t\mathcal {C}^{\alpha }}B_t^{\frac{p+1}{p+2}} \overset{(4.10)}{\lesssim }\,\frac{c^{p+1}}{p} \Vert Z\Vert ^{p+2}_{C_t\mathcal {C}^{\alpha }}+ \frac{1}{pc}B_t. \end{aligned} \end{aligned}$$

(4.18)

For the third term of (4.15), by similar estimates and using that $\mathfrak {m}\in \{0,1\}$ we have,

$$\begin{aligned} \frac{1}{p}|\langle Z_t,w_t^p \mathfrak {m}\rangle | \overset{ (\mathrm{A.2}),\,(4.8)}{\lesssim } \frac{1}{p}\Vert Z\Vert _{C_t\mathcal {C}^\alpha }\mathfrak {m}B^{\frac{p}{p+2}} \overset{(4.10)}{\lesssim } \frac{c^p}{p^2}\Vert Z\Vert ^{p+2}_{C_t\mathcal {C}^\alpha } + \frac{c^p}{p^2}\mathfrak {m}+ \frac{1}{cp}B_t \end{aligned}$$

(4.19)

Regarding the eighth and final term of (4.6),

$$\begin{aligned} \frac{1}{p-1}|\langle Z_t^2,\partial _x \rho _{w_t}\partial _x w_t^{p-1}\rangle |&\overset{(\mathrm{A.2}),\,(\mathrm{A.7})}{\lesssim } \frac{1}{p} \Vert Z\Vert ^2_{C_t\mathcal {C}^{\alpha }}\Vert \partial _x \rho _{w_t}\Vert _{L^\infty }\Vert \partial _x w^{p-1}_t\Vert _{L^1} \nonumber \\&\overset{(4.7),\,(4.9)}{\lesssim }\Vert Z\Vert ^2_{C_t\mathcal {C}^{\alpha }} B_t^{\frac{p}{2(p+2)}}A_t^{\frac{1}{2}} \nonumber \\&\quad \overset{(4.10)}{\le } c\Vert Z\Vert ^4_{C_t\mathcal {C}^{\alpha }} B_t^{\frac{p}{p+2}}+\frac{1}{c}A_t \nonumber \\&\quad \overset{ (4.10)}{\lesssim } \frac{c^{p+1}}{p} \Vert Z\Vert ^{2(p+2)}_{C_t\mathcal {C}^{\alpha }} + \frac{1}{c}B_t + \frac{1}{c}A_t. \end{aligned}$$

(4.20)

So, since $p+1\le \frac{p(1+\alpha )+2+\alpha }{\alpha } =:\beta$ and $c\ge 1$, combining (4.17)–(4.20),

$$\begin{aligned} \sum _{k=1}^2 \left( {\begin{array}{c}2\\ k\end{array}}\right) \frac{1}{p+1-k}|\langle Z_t^k,\partial _x \rho _{w_t}\partial _x w_t^{p+1-k}\rangle | \lesssim \frac{ c^{\beta }}{p}\Vert Z\Vert ^{\frac{p+2}{\alpha }}_{C_t\mathcal {C}^{\alpha }}+\frac{c^{p}}{p^2}\mathfrak {m}+\frac{1}{c}B_t +\frac{1}{c}A_t. \end{aligned}$$

(4.21)

Since $\frac{1}{4(p-1)} \le \frac{3/4}{p+1}$, combining (4.6) with (4.11), (4.14), (4.21) and choosing $c \ge (p\vee \chi )C$ with $C>0$ the sufficiently large implied constant, we obtain that

$$\begin{aligned} \frac{1}{ \chi p(p-1)}\frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}t} \Vert w_t\Vert ^p_{L^p}&\le \max \Big \{\frac{(C (p\vee \chi ))^{\beta }}{p}\Vert Z\Vert ^{\frac{p+2}{\alpha }}_{C_t\mathcal {C}^{\alpha }},\, \frac{(C (p\vee \chi ))^{p+1}\mathfrak {m}}{p^2} \Big \} \\&\quad -\frac{1}{4(p-1)}\Vert w^{p+2}_t\Vert _{L^1}. \end{aligned}$$

By Jensen’s inequality,$(\Vert w_t\Vert ^p_{L^p})^{\frac{p+2}{p}}\le \Vert w^{p+2}_t\Vert _{L^1}$, so that for $C>0$ sufficiently large

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}t} \Vert w_t\Vert ^p_{L^p}+ \frac{\chi p}{4}\left( \Vert w_t\Vert ^p_{L^p}\right) ^{\frac{p+2}{p}} \le \chi \max \Big \{p(C (p\vee \chi ))^{\beta }\Vert Z\Vert ^{\frac{p+2}{\alpha }}_{C_t\mathcal {C}^{\alpha }}, \,(C (p\vee \chi ))^{p+1}\mathfrak {m}\Big \}. \end{aligned}$$

Applying Lemma 4.2 with $f(t)= \Vert w_t\Vert ^p_{L^p}$, $\lambda = \frac{p+2}{p}$ and possibly increasing $C>0$,

$$\begin{aligned} \Vert w_t(\zeta )\Vert ^p_{L^p} \le \max \Big \{ (\chi t/4)^{-\frac{p}{2}}, (C (p\vee \chi ))^{\beta \frac{p}{p+2}}\Vert Z\Vert ^{\frac{p}{\alpha }}_{C_t\mathcal {C}^{\alpha }},\,(C (p\vee \chi ))^{\frac{p(p+1)}{p+2}}\mathfrak {m}/p^{\frac{p}{p+2}}\, \Big \}, \end{aligned}$$

for every $t \in (0,T]$ and $\zeta \in \mathcal {C}^{\alpha _0}$. The stated bound, (4.5), then follows after taking the $p^{\text {th}}$ root on both sides and noticing that $\gamma =\frac{\beta }{p+2}\le \frac{4+3\alpha }{4\alpha }$. $\square$

Remark 4.5

Observe that the bound (4.5) is trivial in the limit $p\rightarrow \infty$, since $\gamma \ge 1$ and so $(p\vee \chi )^{\gamma }\rightarrow \infty$. This is in contrast to [20], where an a priori bound of the same form in $L^\infty (\mathbb {T})$ is shown directly for stochastic reaction-diffusions.

Remark 4.6

In [20], the equivalent of (4.5), in the context of stochastic reaction-diffusion equations, is established with $\Vert Z\Vert ^{1/\alpha }_{C_T\mathcal {C}^{\alpha }}$ replaced by $\Vert Z\Vert ^{1/(1+\alpha )}_{C_T\mathcal {C}^{\alpha }}$. This leads to lighter than Gaussian tail bounds in the case of the reaction-diffusion equation, again contrasted with the heavier than Gaussian tails that we are able to establish for (1.1), see Theorem 1.4.

Remark 4.7

While a mild generalisation of this analysis to f(u), smooth and asymptotically quadratic ($c_1u^2<f(u)<c_2u^2$), would likely be possible, more natural generalisations such as $f(u)=|u|$ or $f(u)=u^{m-1}$ for $m\ge 5$ and odd, seem to present significant challenges. In the case of $f(u)=|u|$, since the testing argument we follow only applies to the remainder $w=u-Z$, the nonlinearity becomes

$$\begin{aligned} \partial _x(|w_t+Z_t|\partial _x\rho _{w_t+Z_t}) = \partial _x (|w_t+Z_t|)\partial _x\rho _{w_t+Z_t} - |w_t+Z_t|(w_T-\bar{w}+Z_t).\end{aligned}$$

When testing with $w_t^{p-1}$ this no longer produces the necessary damping term $-\Vert w_t^{p+2}\Vert _{L^1}$ and so it is not directly clear how one could proceed in this case. Regarding higher polynomial powers, it appears that controlling the transport term causes a problem for $m>3$. In particular estimating the equivalent of (4.15) in the same way, leads us to control $|\langle \partial _x Z , w^{p+m-2}\partial _x \rho _{w}\rangle |$. Repeating similar estimates as in (4.16) and (4.17) lead to the condition $m<\frac{3-\alpha }{1-\alpha }$. For $\alpha \in (0,1/2)$ this restricts us to $m<5$. Letting $\alpha \rightarrow 1$ would allow $m\rightarrow \infty$. Informally speaking, integrability of the solution resulting from the transport term appears to be linked to regularity of the noise - higher regularity leads to higher integrability. This was also observed in the case of stochastic reaction-diffusion equations in [20]. Note also that with $\alpha \ge 1$ one could apply the testing argument directly to u, rather than working with the remainder. In [17] it was in fact shown that for the same model but with $\xi \equiv 0$, global existence and convergence to equilibrium holds for all $m>0$. In the case of more regular noise $v\in C_T\mathcal {C}^{1}(\mathbb {T})$, we would therefore expect at least an analogue of Theorem 4.4 to hold directly for u also for all $m>0$.

Proposition 4.8

Let $\mathfrak {m}\in \{0,1\}$, $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$ and $Z \in C_T\mathcal {C}^{\alpha }_0$. Then there exists a unique mild solution $w(\zeta )\in C_{\eta ;T}\mathcal {C}^{\alpha }$ to (3.1). Furthermore, the map,

$$\begin{aligned} \begin{aligned} \mathscr {S}: \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})\times C_T \mathcal {C}^{\alpha }_0(\mathbb {T})&\rightarrow C_{\eta ;T} \mathcal {C}^{\alpha }_\mathfrak {m}(\mathbb {T})\\ (\zeta ,Z)&\mapsto u(\zeta ):=w(\zeta )+Z, \end{aligned} \end{aligned}$$

(4.22)

is jointly, locally Lipschitz.

Proof

We recall from Theorem 3.4 that there exists a $T_*\in (0,1)$, depending only on $\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}$ and $\Vert Z\Vert _{C_T\mathcal {C}^\alpha }$, such that a mild solution $w \in C_{\eta ;T_*}\mathcal {C}^{\alpha }$ exists to (3.1). Without loss of generality let us assume $T>T_*$ and that we fix an even $p \in (-\frac{1}{\alpha _0},\infty )$ so that $L^p(\mathbb {T})\hookrightarrow \mathcal {C}^{\alpha _0}(\mathbb {T})$. In this case, it is clear that we can extend the solution for a positive time of existence so long as $\Vert w_t\Vert _{L^p}$ remains finite. However, by Theorem 4.4, $\Vert w_t\Vert _{L^p}$ is bounded above by a function of t independent of the initial data and so we may continue the solution to all of [0, T]. From Corollary 3.7, for $u_t:= w_t+Z_t$, we have $\bar{u}_t = \bar{\zeta }+\bar{Z}_t$, for all $t\in (0,T]$. Similarly, for all $t>0$, $\Vert u_t\Vert _{\mathcal {C}^{\alpha }}\le \Vert w_t\Vert _{\mathcal {C}^{\alpha }}+\Vert Z_t\Vert _{\mathcal {C}^{\alpha }}$. Hence the solution map (4.22) is well-defined and $\Vert u\Vert _{C_{\eta ,T}\mathcal {C}^\alpha }$ depends only on $\Vert Z\Vert _{C_T\mathcal {C}^\alpha }$.

To continue the proof, we state the following

Lemma 4.9

Consider $\mathfrak {R}>0$ and define the set

$$\begin{aligned} D_\mathfrak {R}:= \{(\zeta ,Z)\in \mathcal {C}^{\alpha _0}_\mathfrak {m}\times C_T\mathcal {C}^\alpha \,:\,\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}} + \Vert Z\Vert _{C_T\mathcal {C}^\alpha } < \mathfrak {R}\}. \end{aligned}$$

Then for $T_*=T_*(\mathfrak {R})>0$ sufficiently small, $\mathscr {S}:D_\mathfrak {R}\rightarrow C_\eta ((0,2T_*],\mathcal {C}^\alpha _\mathfrak {m})$ is K-Lipschitz where $K=K(\mathfrak {R})>0$.

Proof

Let $(\zeta ,Z),(\tilde{\zeta },\tilde{Z})\in D_\mathfrak {R}$ and consider the corresponding solutions

$$\begin{aligned} u_t&= e^{t\Delta }\zeta + \int _0^t e^{(t-s)\Delta }\chi \partial _x(u^2_s\partial _x\rho _{u_s})\,\mathop {}\!\mathrm {d}s + Z_t,\\ \tilde{u}_t&= e^{t\Delta }\tilde{\zeta } + \int _0^t e^{(t-s)\Delta }\chi \partial _x(\tilde{u}^2_s\partial _x\rho _{\tilde{u}_s})\,\mathop {}\!\mathrm {d}s + \tilde{Z}_t. \end{aligned}$$

From Theorem 3.4, there exists $T_*(\mathfrak {R})>0$ such that $\Vert u\Vert _{C_{\eta ;2T_*}\mathcal {C}^{\alpha }}\vee \, \Vert \tilde{u}\Vert _{C_{\eta ;2T_\star }\mathcal {C}^{\alpha }}\le 2$. For $t \in (0,2T_*]$, using Theorem A.6 and similar bounds as in the proof of Theorem 3.4 we see that

$$\begin{aligned} \Vert u_t-\tilde{u}_t\Vert _{\mathcal {C}^{\alpha }}&\lesssim t^{-\frac{\alpha -\alpha _0}{2}}\Vert \zeta -\tilde{\zeta }\Vert _{\mathcal {C}^{\alpha _0}} + t^{\frac{1}{2}-3\eta } \Vert u-\tilde{u}\Vert _{C_{\eta ;t}\mathcal {C}^{\alpha }}+ \Vert Z_t-\tilde{Z}_t\Vert _{\mathcal {C}^{\alpha }}, \end{aligned}$$

where the proportionality constant does not depend on $\zeta ,\tilde{\zeta },Z,\tilde{Z}$. So multiplying through by $t^\eta$ and taking the supremum over $t\in (0,2T_*]$ we have that

$$\begin{aligned} \Vert u-\tilde{u}\Vert _{C_{\eta ;2T_*}\mathcal {C}^{\alpha }} \lesssim \Vert \zeta -\tilde{\zeta }\Vert _{\mathcal {C}^{\alpha _0}} + T_*^{\frac{1}{2}-2\eta } \Vert u-\tilde{u}\Vert _{C_{\eta ;S}\mathcal {C}^{\alpha }} + \Vert Z-\tilde{Z}\Vert _{C_T\mathcal {C}^{\alpha }}. \end{aligned}$$

By lowering $T_*$ further, we obtain

$$\begin{aligned} \Vert u-\tilde{u}\Vert _{C_{\eta ;2T_*}\mathcal {C}^{\alpha }} \lesssim \Vert \zeta -\tilde{\zeta }\Vert _{\mathcal {C}^{\alpha _0}} +\Vert Z-\tilde{Z}\Vert _{C_T\mathcal {C}^{\alpha }}. \end{aligned}$$

$\square$

We now prove that $\mathscr {S}$ is jointly locally Lipschitz. Consider $\mathfrak {R}>1$, $D_\mathfrak {R}$, and $T_*(\mathfrak {R})>0$ as in Lemma 4.9. Then $\mathscr {S}$ is $K(\mathfrak {R})$-Lipschitz from $D_\mathfrak {R}$ to $C_\eta ((0,2T_*],\mathcal {C}^{\alpha })$. In particular, the map $(\zeta ,Z) \mapsto u|_{[T_*,2T_*]} \in C([T_*,2T_*],\mathcal {C}^\alpha )$ is $K(\mathfrak {R})$-Lipschitz. If $T\le 2T_*$, then we are done. Hence, suppose $T>2T_*$. We will prove that, for $\bar{T}_*(\mathfrak {R})\in (0,T_*]$ sufficiently small, $u|_{[T_*+\bar{T}_*,T]}\in C([\bar{T}_*,T],\mathcal {C}^\alpha )$ is a Lipschitz function of $(\zeta ,Z)$, from which the conclusion will follow.

To this end, consider $(\zeta ,Z)\in D_\mathfrak {R}$ and let u be the corresponding solution. By the a priori estimate, for $p\ge 2$ sufficiently large $\Vert u_t\Vert _{\mathcal {C}^{\alpha _0}}\lesssim _{\alpha _0,p} \Vert u_t\Vert _{L^p}\lesssim t^{-1/2}\vee \mathfrak {R}^{1/\alpha }$ for all $t>0$. Therefore, there exists $\bar{\mathfrak {R}}(\mathfrak {R})>0$ such that $\Vert u_t\Vert _{\mathcal {C}^{\alpha _0}}\le \bar{\mathfrak {R}}$ for all $t\in [T_*,T]$.

Let $\bar{T}_*(\bar{\mathfrak {R}})>0$ be the constant from Lemma 4.9. Without loss of generality, we can assume that $\bar{T}_*\le T_*$ and that $T=T_*+N \bar{T}_*$ for some integer $N\ge 2$. By the above remark, for all $1\le n < N$, $u|_{[T_*+n \bar{T}_*,T_*+(n+1)\bar{T}_*]}\in C([T_*+n\bar{T}_*,T_*+(n+1)\bar{T}_*],\mathcal {C}^\alpha )$ is a $K(\mathfrak {R})$-Lipschitz function of $(u_{T_*+(n-1)\bar{T}_*},Z)\in \mathcal {C}^{\alpha _0}\times C_T\mathcal {C}^\alpha$. Combining these estimates together, it follows that $u|_{[T_*+\bar{T}_*,T]}\in C([\bar{T}_*,T],\mathcal {C}^\alpha )$ is a Lipschitz function of $(\zeta ,Z)$. $\square$

We conclude this section with the proof of Theorem 1.1.

Proof of Theorem 1.1

From Theorem 2.3, for any $T>0$ and $\alpha <\frac{1}{2}$, $v_{0,\,\cdot \,}\in C_T\mathcal {C}_0^{\alpha }$ (in fact $v_{0,\,\cdot \,} \in \cap _{\kappa \in [0,1/2)}\mathcal {C}^{\kappa }_T\mathcal {C}_0^{\alpha -2\kappa }$). Hence Proposition 4.8 implies that $\mathbb {P}$-a.s. there exists a unique mild solution $u(\zeta ):=w(\zeta )+v_{0,\,\cdot \,}=\mathscr {S}(\zeta ,v_{0,\,\cdot \,})$ to (1.1) on [0, T]. $\square$

5 Invariant measure

Having established global well-posedness of the SPDE (1.1) we turn to study its long time behaviour. In this section we prove Theorem 1.4 which implies existence and uniqueness of the invariant measure and demonstrates exponential convergence of the adjoint semi-group to the invariant measure in the topology of total variation.

By Remark 4.3, appropriately adjusting the spatial mean $\mathfrak {m}$ of the initial condition $\zeta$, it suffices to prove Theorem 1.4 in the case $\chi =1$. We thus assume $\chi =1$ throughout this section.

Definition 5.1

Let $v_{t_0,\,\cdot \,}$ denote the mild solution to the stochastic heat equation as in Definition 2.2 and denote $v_t:=v_{0,t}$. For $\zeta \in \mathcal {C}^{\alpha _0}$, the process $u(\zeta ):=w(\zeta )+v_{\cdot }=\mathscr {S}(\zeta ,v_{\cdot })$, with $\mathscr {S}$ as defined in Proposition 4.8, is called the mild solution on [0, T] to (1.1) with initial condition $\zeta$.

We define the semi-group associated to (1.1) by setting, for all $t\ge 0$,

$$\begin{aligned} P_t\Phi (\zeta ):= \mathbb {E}\left[ \Phi (u_t(\zeta ))\right] , \quad \text { for all } \Phi \in \mathcal {B}_b(\mathcal {C}^{\alpha _0}(\mathbb {T})). \end{aligned}$$

We first show that $(u_t)_{t>0}$ is a Markov process with Feller semi-group $(P_t)_{t>0}$ and that for each $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$, there exists a measure $\nu _\zeta \in \mathcal {P}(\mathcal {C}^{\alpha _0}_\mathfrak {m})$, invariant for $(P_t)_{t>0}$.

Lemma 5.2

Let u be a mild solution to (1.1). Then for every $t \in [0,T]$ and $h \in (0,T-t)$ we have the identity

$$\begin{aligned} u_{t+h} = \tilde{w}_{t,t+h}+v_{t,t+h} , \end{aligned}$$

(5.1)

where $\tilde{w}_{t,t+h}$ solves

$$\begin{aligned} \tilde{w}_{t,t+h} = e^{h\Delta }u_t + \int _0^h e^{(h-r)\Delta }\partial _x ((\tilde{w}_{t,t+r}+v_{t,t+r})^2\partial _x \rho _{\tilde{w}_{t,t+r}+v_{t,t+r}})\,\mathop {}\!\mathrm {d}r. \end{aligned}$$

(5.2)

Proof

A simple consequence of the heat semi-group property.

Theorem 5.3

For $\mathfrak {m}\in \mathbb {R}$, and $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$, let $u(\zeta )\in C_{\eta ;T}\mathcal {C}^{\alpha }$ be the unique solution to (1.1) as in Definition 5.1. Then for every $p\in [1,\infty )$, we have that

$$\begin{aligned} \sup _{\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})}\sup _{T>0} \sup _{t\in (0,T]} \left( t^{\frac{p}{2}}\wedge 1 \right) \mathbb {E}\left[ \Vert u_t(\zeta )\Vert ^p_{L^{p}} \right] <\infty . \end{aligned}$$

(5.3)

Proof

It suffices to consider $T>1$. Consider first $p\ge 2$ even. For $t \in (0,1]$, applying (4.5) and Theorem 2.3,

$$\begin{aligned} \mathbb {E}\left[ \Vert u_t\Vert ^p_{L^p}\right] \lesssim \mathbb {E}\left[ \Vert w_t\Vert _{L^p}^p \right] + \mathbb {E}\left[ \Vert v_t\Vert _{L^p}^p\right] \lesssim t^{-\frac{p}{2}}+1. \end{aligned}$$

(5.4)

For $t\in (1,T]$ we employ the Markov decomposition (5.1) to give

$$\begin{aligned} \mathbb {E}\left[ \Vert u_t\Vert _{L^p}^p\right] \lesssim \mathbb {E}\left[ \Vert \tilde{w}_{t-1,t}\Vert _{L^p}^p\right] + \mathbb {E}\left[ \Vert v_{t-1,t}\Vert _{L^p}^p \right] . \end{aligned}$$

Then, observing that $\tilde{w}_{t-1,t}$ solves (3.1) with initial condition $u_{t-1}(\zeta )$ and driving path $Z_t=v_{t-1,t}$, by Theorem 4.4 we have

$$\begin{aligned} \mathbb {E}\left[ \Vert \tilde{w}_{t-1,t}\Vert _{L^p}^p \right] \le C_{p,\mathfrak {m}} \max \Big \{ \Vert v_{t-1,\,\cdot \,}\Vert ^{\frac{p}{\alpha }}_{C_{[t-1,t]}\mathcal {C}^{\alpha }},\,1\,\Big \}. \end{aligned}$$

So, since the law of $v_{t-1,t}$ does not depend on $t>1$, we have

$$\begin{aligned} \sup _{\zeta \in \mathcal {C}^{\alpha _0}_{\mathfrak {m}}(\mathbb {T})}\sup _{T>0} \sup _{t\in (1,T]} \mathbb {E}\left[ \Vert u_t(\zeta )\Vert _{L^p}^p\right] <\infty . \end{aligned}$$

(5.5)

Combining (5.4) with (5.5) proves (5.3) for even $p\ge 2$. The case of general $p\ge 1$ follows by Jensen’s inequality. $\square$

5.1 Existence of invariant measures

We use the decomposition (5.1) to show that $t\mapsto u_t$ defines a Markov process.

Theorem 5.4

Let $\zeta \in \mathcal {C}^{\alpha _0}(\mathbb {T})$ and $u_t(\zeta )$ as in Definition 5.1. Then for any $\Phi \in \mathcal {B}_b(\mathcal {C}^{\alpha _0}(\mathbb {T}))$

$$\begin{aligned} \mathbb {E}\left[ \Phi (u_{t+h}(\zeta ))\,\big |\, \mathcal {F}_t \right] = P_h(\Phi (u_{t}(\zeta )). \end{aligned}$$

In particular, $t\mapsto u_t$ is a Markov process with associated Markov semi-group $(P_t)_{t>0}$.

Proof

Using the Markov decomposition (5.1) we see that

$$\begin{aligned} \Phi (u_{t+h}(\zeta )) = \Phi (\tilde{w}_{t,t+h}(u_t(\zeta ))+v_{t,t+h})=\Phi (u_{t,t+h}(u_t(\zeta ))), \end{aligned}$$

where $\tilde{w}_{t,t+h}(u_t(\zeta ))$ solves (5.2). We define

$$\begin{aligned} \psi (u_t(\zeta ),v_{t,t+h}) := \Phi (\tilde{w}_{t,t+h}(u_t(\zeta ))+v_{t,t+h}) \end{aligned}$$

and then since $u_t(\zeta )$ is $\mathcal {F}_t$ measurable and $v_{t,t+h}\perp \mathcal {F}_t$, applying [6, Prop. 1.12], we have that

$$\begin{aligned} \mathbb {E}\left[ \Phi (u_{t+h}(\zeta ))\,\big |\, \mathcal {F}_t \right] = \mathbb {E}[\Phi (u_{t,t+h}(u_t(\zeta )))] = P_h(\Phi (u_t(\zeta ))) \end{aligned}$$

and so the claim is shown. $\square$

The following lemma shows that $(P_t)_{t>0}$ is a Feller semi-group in the sense of [5, Sec. 3.1].

Lemma 5.5

For any $\Phi \in \mathcal {C}_b(\mathcal {C}^{\alpha _0}(\mathbb {T}))$ and $t> 0$ we have that $P_t \Phi \in \mathcal {C}_b(\mathcal {C}^{\alpha _0}(\mathbb {T}))$ and for any $\zeta \in \mathcal {C}^{\alpha _0}$, $\lim _{t\rightarrow 0}|P_t\Phi (\zeta ) -\Phi (\zeta )|=0$.

Proof

By Proposition 4.8, $\mathcal {C}^{\alpha _0}(\mathbb {T})\ni \zeta \mapsto u_t(\zeta )\in \mathcal {C}^{\alpha }(\mathbb {T})$ is $\mathbb {P}$-a.s. continuous for every $t>0$. The fact that $P_t\Phi \in \mathcal {C}_b(\mathcal {C}^{\alpha _0}(\mathbb {T}))$ thus follows from the dominated convergence theorem. The fact that $\lim _{t\searrow 0}|P_t\Phi (\zeta )-\Phi (\zeta )|= 0$ for all $\zeta \in \mathcal {C}^{\alpha _0},\,\Phi \in \mathcal {C}_b(\mathcal {C}^{\alpha _0}(\mathbb {T}))$, similarly follows the dominated convergence theorem and (3.9). $\square$

We are now in a position to show the existence of an invariant measure, $\nu _\zeta \in \mathcal {P}(\mathcal {C}^{\alpha _0}_\mathfrak {m})$ for every $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$.

Theorem 5.6

Let $\mathfrak {m}\in \mathbb {R}$. Then for every $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$, there exists a measure $\nu _\zeta \in \mathcal {P}(\mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T}))$ and an increasing sequence of times $t_k \nearrow \infty$ such that,

$$\begin{aligned} \frac{1}{t_k}\int _0^{t_k} P^*_r \,\delta _\zeta \,\mathop {}\!\mathrm {d}r \rightharpoonup \nu _\zeta , \quad \text { as } k\rightarrow \infty . \end{aligned}$$

Furthermore $\nu _\zeta$ is invariant for $(P_t)_{t>0}$.

Proof

By the compact embedding $L^p\hookrightarrow \mathcal {C}^{\alpha _0}$ for sufficiently large $p\ge 1$, it follows from Theorem 5.3 that the family of measures $\{P^*_t \delta _\zeta \}_{t\ge 1, \zeta \in \mathcal {C}^{\alpha _0}(\mathbb {T})}$ is tight. In particular, for every sequence of times $(t_k)_{k>0}$, $\left( R_{t_k}^* \delta _\zeta \right) _{k>0} := \left( \frac{1}{t_k}\int _0^{t_k} P^*_r \,\delta _\zeta \,\mathop {}\!\mathrm {d}r\right) _{k>0}$ is a tight family of measures. Applying Prokhorov’s theorem gives the existence of a sequence $(t_k)_{k>0}$ for which the necessary weak convergence holds. From [5, Thm. 3.1.1] we see that the limit measure is necessarily invariant for $(P_t)_{t> 0}$. $\square$

5.2 Strong feller property

We prove that $(P_t)_{t>0}$ possesses the strong Feller property closely following the method employed in [34]. The main step is to establish a Bismut–Elworthy–Li (BEL) type formula, similar to that given in [34, Thm. 5.5].

For technical reasons in this section we replace (1.3) with the assumption that $\alpha _0\in \left( -\frac{1}{3},0\right)$, $\alpha \in \left( 0,\alpha _0+\frac{1}{3}\right)$ and $\eta >0$ such that,

$$\begin{aligned} \frac{\alpha -\alpha _0}{2}<\eta < \frac{1}{6}. \end{aligned}$$

(5.6)

We note that all previous results also hold for this more restrictive parameter range and this restriction does not affect the main result as stated, see the proof of Theorem 1.4 at the conclusion of Sect. 5.4. Let $(e_m)_{m\in \mathbb {Z}}$, $e_m(x)=e^{i2\pi mx}$, be the usual Fourier basis elements of $L^2(\mathbb {T})$ and $(\Delta _k)_{k\ge -1}$ be the Littlewood–Paley projection operators, see “Appendix A” for more details. For $\varepsilon \in (0,1)$, we define $\Pi _\varepsilon (L^2(\mathbb {T}))$ as the space of real functions spanned by $\left\{ (e_m)_{|m|<\frac{1}{\varepsilon }}\right\}$ and

$$\begin{aligned} \hat{\Pi }_\varepsilon := \sum _{ -1\le k \le -\log _{2}(9\varepsilon )} \Delta _k :L^2(\mathbb {T}) \rightarrow \Pi _\varepsilon (L^2(\mathbb {T}^d)). \end{aligned}$$

Observe that there exists $\ell _\varepsilon :\mathbb {Z}\rightarrow [0,1]$ with $\ell _\varepsilon (m)=0$ for $|m|\ge \frac{1}{\varepsilon }$ such that $\mathcal {F}(\hat{\Pi }_\varepsilon f)(m) = \ell _\varepsilon (m) \mathcal {F}f(m)$ where $\mathcal {F}$ is the Fourier transform. Furthermore (see e.g. [34, p. 1213]),

• i) $\displaystyle \Vert \hat{\Pi }_\varepsilon \Vert _{\text {Op}(\mathcal {C}^\beta ;\,\mathcal {C}^{\beta })} :=\sup _{\Vert u\Vert _{\mathcal {C}^{\beta }}\le 1 }\Vert \hat{\Pi }_\varepsilon u\Vert _{\mathcal {C}^{\beta }} \le 1$ for all $\varepsilon \in (0,1)$ and $\beta \in \mathbb {R}$,

• ii) for every $\beta \in \mathbb {R}$ and $\delta >0$, there exists $C>0$ such that for all $\varepsilon \in (0,1)$

  $$\begin{aligned} \Vert \hat{\Pi }_\varepsilon f-f\Vert _{\mathcal {C}^{\beta -\delta }} \le C \varepsilon ^\delta \Vert f\Vert _{\mathcal {C}^{\beta }}. \end{aligned}$$

We fix for the rest of the subsection $\delta >0$

$$\begin{aligned} \alpha +\delta <\frac{1}{2},\quad \eta -\frac{\alpha -\alpha _0+\delta }{2}>0,\quad \frac{1-\delta }{2}-3\eta >0. \end{aligned}$$

(5.7)

Such a $\delta$ exists due to (5.6).

Consider now $Z\in C_T\mathcal {C}^{\alpha +\delta }$ and let $Z_{\varepsilon ;t} := \hat{\Pi }_\varepsilon Z_t$. Consider likewise $\zeta \in \mathcal {C}^{\alpha _0}$ and let $\zeta _\varepsilon :=\hat{\Pi }_\varepsilon \zeta$. We define a smooth approximation to (1.1), with $\chi =1$ and deterministic noise, by

$$\begin{aligned} u_{\varepsilon ;t} = e^{t\Delta }\zeta _\varepsilon + \int _0^t e^{(t-s)\Delta } \partial _x\hat{\Pi }_\varepsilon (u^2_{\varepsilon ;s}\partial _x\rho _{u_{\varepsilon ;s}})\mathop {}\!\mathrm {d}s + Z_{\varepsilon ;t}, \end{aligned}$$

(5.8)

where, as usual,

$$\begin{aligned} -\partial _{xx} \rho _{ u_{\varepsilon ;t}} =u_{\varepsilon ;t}-\bar{u}_\varepsilon , \text { on }\mathbb {T}. \end{aligned}$$

Remark 5.7

We will later set, right before Theorem 5.13, $Z_\varepsilon =\hat{\Pi }_\varepsilon v$, where v is the SHE.

Theorem 5.8

Let $T>0$, $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$, and $Z\in C_T\mathcal {C}^{\alpha +\delta }$. For every $\bar{\varepsilon }>0$, there exists $\varepsilon _0(\bar{\varepsilon },T,\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}+\Vert Z\Vert _{C_T\mathcal {C}^{\alpha +\delta }})>0$ such that, for every $\varepsilon \in (0,\varepsilon _0)$ there exists a unique solution $u_\varepsilon \in C_{T}\Pi _\varepsilon (L^2(\mathbb {T}))$ to (5.8). Furthermore

$$\begin{aligned} \Vert u_{t}-u_{\varepsilon ;t}\Vert _{C_{\eta ;T}\mathcal {C}^\alpha } \le \bar{\varepsilon }. \end{aligned}$$

(5.9)

Proof

It follows from Properties i) and ii) of the projection and the same steps as in the proof of Theorem 3.4 that for every $\varepsilon >0$ there exists $T_\varepsilon (\Vert \zeta _\varepsilon \Vert _{\mathcal {C}^{\alpha _0}}+\Vert Z_\varepsilon \Vert _{C_T\mathcal {C}^\alpha })>0$, such that $u_\varepsilon \in C_{\eta ;T_\varepsilon }\mathcal {C}^\alpha$ solves (5.8) and satisfies $\Vert u_\varepsilon \Vert _{C_{\eta ;T_\varepsilon }\mathcal {C}^\alpha } \le 1$. The fact that $u_\varepsilon \in C_{T_\varepsilon }\Pi _\varepsilon (L^2(\mathbb {T}))$ follows from the same argument as the end of Theorem 3.4 since $\zeta _\varepsilon \in C^\infty$.

To continue, we state the following

Lemma 5.9

Consider $\mathfrak {R}>0$ and define $D_\mathfrak {R}$ as in Lemma 4.9. There exists $T_*(\mathfrak {R})>0$ such that, if $(\zeta _\varepsilon ,Z_\varepsilon ),(\tilde{\zeta },\tilde{Z})\in D_\mathfrak {R}$, then

$$\begin{aligned} \Vert \tilde{u}_t-u_{\varepsilon ;t}\Vert _{C_{\eta ;T_*}\mathcal {C}^{\alpha }} \le \Vert \zeta _\varepsilon -\tilde{\zeta }\Vert _{\mathcal {C}^{\alpha _0-\delta }} + \varepsilon ^{\delta } +\Vert \tilde{ Z}-Z_{\varepsilon }\Vert _{C_T\mathcal {C}^{\alpha }}, \end{aligned}$$

where $\tilde{u}=\mathscr {S}(\tilde{\zeta },\tilde{Z})$ with $\mathscr {S}$ as in Proposition 4.8.

Proof

There exists $T_*(\mathfrak {R})>0$ such that $\Vert \tilde{u}\Vert _{C_{\eta ;2T_*}\mathcal {C}^{\alpha }}+\Vert u_\varepsilon \Vert _{C_{\eta ;2T_\star }\mathcal {C}^{\alpha }}\le 2$. For $t \in (0,2T_*]$,

$$\begin{aligned} \Vert \tilde{u}_t-u_{\varepsilon ;t} \Vert _{\mathcal {C}^{\alpha }}&\le \Vert e^{t\Delta }(\tilde{\zeta }-\zeta _\varepsilon )\Vert _{\mathcal {C}^{\alpha }} + \int _0^t \Vert e^{(t-s)\Delta } \partial _x (\tilde{ u}^2_s\partial _x \rho _{\bar{u}_s}\\&\quad - \hat{\Pi }_\varepsilon (u^{2}_{\varepsilon ;s} \partial _x\rho _{u_{\varepsilon ;s}}))\Vert _{\mathcal {C}^{\alpha }}\,\mathop {}\!\mathrm {d}s\\&\quad + \Vert \tilde{ Z}_t-Z_{\varepsilon ;t}\Vert _{\mathcal {C}^{\alpha }}. \end{aligned}$$

Then

$$\begin{aligned} \Vert e^{t\Delta }(\tilde{\zeta }-\zeta _\varepsilon ) \Vert _{\mathcal {C}^{\alpha }} \lesssim t^{-\frac{\alpha -\alpha _0+\delta }{2}}\Vert \tilde{\zeta }-\zeta _\varepsilon \Vert _{\mathcal {C}^{\alpha _0-\delta }}, \end{aligned}$$

and applying the triangle inequality and using Property ii) of the projection,

$$\begin{aligned} \Vert \tilde{u}_s^2\partial _x \rho _{\bar{u}_s} - \hat{\Pi }_\varepsilon (u^2_{\varepsilon ;s}\partial _x \rho _{u_{\varepsilon ;s}})\Vert _{\mathcal {C}^{\alpha -\delta }}&\le \Vert \tilde{u}_s^2\partial _x \rho _{\tilde{u}_s} -u^2_{\varepsilon ;s}\partial _x \rho _{u_{\varepsilon ;s}} \Vert _{\mathcal {C}^{\alpha }}\\&\quad + \Vert u^2_{\varepsilon ;s}\partial _x \rho _{u_{\varepsilon ;s}}-\hat{\Pi }_\varepsilon (u^2_{\varepsilon ;s}\partial _x \rho _{u_{\varepsilon ;s}})\Vert _{\mathcal {C}^{\alpha -\delta }}\\&\lesssim s^{-2\eta }\Vert \tilde{u}_s-u_{\varepsilon ;s}\Vert _{\mathcal {C}^{\alpha }} + \varepsilon ^{\delta }s^{-3\eta }\\&\le s^{-3\eta }\Vert \tilde{u}-u_\varepsilon \Vert _{C_{\eta ;s}\mathcal {C}^\alpha } +\varepsilon ^{\delta }s^{-3\eta }. \end{aligned}$$

Therefore, using that $\eta <\frac{1}{6}$, we may integrate $\Vert e^{(t-s)\Delta } \partial _x (\tilde{ u}^2_s\partial _x \rho _{\bar{u}_s} - \hat{\Pi }_\varepsilon (u^{2}_{\varepsilon ;s} \partial _x \rho _{u_{\varepsilon ;s}}))\Vert _{\mathcal {C}^{\alpha }}$ from 0 to t and take suprema to obtain

$$\begin{aligned} \sup _{t \in (0,T_*]}t^{\eta }\Vert \tilde{u}_s-u_{\varepsilon ;s}\Vert _{\mathcal {C}^{\alpha }}&\lesssim T_*^{\eta -\frac{\alpha -\alpha _0-\delta }{2}}\Vert \tilde{\zeta }-\zeta _\varepsilon \Vert _{\mathcal {C}^{\alpha _0-\delta }} + T_*^{\frac{1-\delta }{2}-3\eta } \sup _{t \in (0,T_*]}t^{\eta }\Vert \tilde{u}_t-u_{\varepsilon ;t}\Vert _{\mathcal {C}^{\alpha }}\\&\quad + T_*^{\frac{1-\delta }{2}-2\eta }\varepsilon ^{\delta } +T_*^{\eta }\Vert \tilde{Z}-Z_{\varepsilon }\Vert _{C_T\mathcal {C}^{\alpha }}. \end{aligned}$$

After potentially lowering $T_*(\mathfrak {R})>0$ further, we obtain

$$\begin{aligned} \sup _{t \in (0,T_*]}t^\eta \Vert \tilde{u}_t-u_{\varepsilon ;t}\Vert _{\mathcal {C}^{\alpha }} \le \Vert \tilde{\zeta }-\zeta _\varepsilon \Vert _{\mathcal {C}^{\alpha _0-\delta }} + \varepsilon ^{\delta } +\Vert \tilde{Z}-Z_{\varepsilon }\Vert _{C_T\mathcal {C}^{\alpha }} . \end{aligned}$$

$\square$

To conclude the proof of Theorem 5.8, observe that $\Vert \zeta -\Pi _\varepsilon \zeta \Vert _{\mathcal {C}^{\alpha _0-\delta }} \lesssim \varepsilon ^\delta \Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}$ and $\Vert Z_t-Z_{\varepsilon ;t}\Vert _{\mathcal {C}^{\alpha }} \lesssim \varepsilon ^\delta \Vert Z_t\Vert _{\mathcal {C}^{\alpha +\delta }}$ by Property ii). By the a priori estimate, for every $\mathfrak {R}>0$ there exists $T_*(\mathfrak {R})>0$ as in Lemma 5.9 and $\bar{\mathfrak {R}}(\mathfrak {R})>0$ such that $\Vert u_t\Vert _{\mathcal {C}^{\alpha _0}}\le \bar{\mathfrak {R}}$ for all $t\in [T_*,T]$. Iteratively applying Lemma 5.9 as in the proof of Proposition 4.8, there exists $\varepsilon _0(\mathfrak {R})>0$ sufficiently small such that $\Vert u_{\varepsilon ;t}\Vert _{\mathcal {C}^\alpha } \le 2\bar{\mathfrak {R}}$ on $[\bar{T}_*,T]$ for all $\varepsilon <\varepsilon _0$, from which (5.9) follows after another iterative application of Lemma 5.9. $\square$

By Theorem 5.8, for every $\mathfrak {R}>0$ and $T>0$, there exists $\varepsilon _0(\mathfrak {R},T)>0$ sufficiently small such that for all $\varepsilon \in (0,\varepsilon _0)$ the solution map to (5.8)

$$\begin{aligned} \begin{aligned} \mathscr {S}^\varepsilon :A_\mathfrak {R}&\rightarrow C_{T}\Pi _\varepsilon (L^2(\mathbb {T}))\\ \left( \zeta ,Z\right)&\mapsto u_{\varepsilon }, \end{aligned} \end{aligned}$$

(5.10)

is well-defined and $\Vert u_\varepsilon -u\Vert _{C_{\eta ;T}\mathcal {C}^\alpha }\le 1$, where $u=\mathscr {S}(\zeta ,Z)$ and $A_\mathfrak {R}:= \{(\zeta , Z\} \in \mathcal {C}^{\alpha _0}\times C_T\mathcal {C}^{\alpha +\delta }\,:\,\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}+\Vert Z\Vert _{C_T\mathcal {C}^{\alpha +\delta }}<\mathfrak {R}\}$.

We now fix for the remainder of the section $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}$, $T>0$, and $Z\in C_T\mathcal {C}^{\alpha +\delta }$ with $Z_0=0$. We let $\varepsilon _0>0$ sufficiently small such that $\mathscr {S}^\varepsilon (\zeta ,Z)$ is well-defined and $\Vert u_\varepsilon -u\Vert _{C_{\eta ;T}\mathcal {C}^\alpha }\le 1$ for all $\varepsilon \in (0,\varepsilon _0)$. In the sequel we will always let $\varepsilon \in (0,\varepsilon _0)$.

We define $N_\varepsilon :=\{m\in \mathbb {Z}\,:\,|m|<\frac{1}{\varepsilon },\;m\ne 0 \}$ and further assume that $Z_{\varepsilon ;t}:=\hat{\Pi }_\varepsilon Z_{t}$ is of the form

$$\begin{aligned} Z_{\varepsilon ;t} =\sum _{m\in N_\varepsilon } \int _0^t\ell _\varepsilon (m) e^{-4\pi ^2|m|^2(t-s)}\mathop {}\!\mathrm {d}\hat{B}_{m;s} \,e_m, \end{aligned}$$

(5.11)

where $\{\hat{B}_m\}_{m\in \mathbb {Z},m\ne 0}$ are functions $\hat{B}_m\in C([0,T],\mathbb {C})$ which satisfy the reality condition

$$\begin{aligned} \hat{B}_{-m} = \overline{\hat{B}_m}. \end{aligned}$$

(5.12)

We denote

$$\begin{aligned} \hat{B}_\varepsilon := (\hat{B}_m)_{m\in N_\varepsilon } \in \bar{C}_T\mathbb {C}^{N_\varepsilon }, \end{aligned}$$

where $\bar{C}_T\mathbb {C}^{N_\varepsilon }\subset C_T\mathbb {C}^{N_\varepsilon }$ is the subspace of $(\hat{B}_m)_{m\in N_\varepsilon }$ which satisfies the reality condition (5.12).

Remark 5.10

We will henceforth identify $\bar{C}_T\mathbb {C}^{N_\varepsilon }$ with a subspace of $C_T C^{\infty }_0\subset C_T\mathcal {C}^{\alpha +\delta }_0$ by mapping $(\hat{B}_m)_{m\in N_\varepsilon }$ to $Z_\varepsilon$ via (5.11). Observe that the integral in (5.11) is well-defined as a Riemann–Stieltjes integral for any $\hat{B} \in \bar{C}_T\mathbb {C}^{N_\varepsilon }$. For fixed $\zeta \in \mathcal {C}^{\alpha _0}$, we will treat in this way $u_\varepsilon (\zeta ,Z):=\mathscr {S}^\varepsilon (\zeta ,Z)$ as a function of $\hat{B}_\varepsilon$ whenever it is well-defined.

We now prove the differentiability of $\mathscr {S}^\varepsilon$ with respect to both arguments separately, using $\mathcal {D}$ for derivatives with respect to $\hat{B}_\varepsilon$ and D for derivatives with respect to $\zeta$. For $\mathfrak {R}\ge 1$, we recall the definition of $T_*(\mathfrak {R})$ given by (3.7).

Lemma 5.11

(Derivative in Noise) There exists an open neighbourhood $\mathcal {O}_{\hat{B}_\varepsilon } \subset \bar{C}_T\mathbb {C}^{N_\varepsilon }$ containing $\hat{B}_\varepsilon$ such that $u_\varepsilon (\zeta ,\cdot )$ is Fréchet differentiable as a mapping from $\mathcal {O}_{\hat{B}_\varepsilon }$ to $C_{T}\Pi _\varepsilon L^2(\mathbb {T})$. Furthermore, for any $f \in \bar{C}_T\mathbb {C}^{N_\varepsilon }$, such that $f|_{t=0}=0$, the directional derivative $\mathcal {D}_f u_\varepsilon$ satisfies the equation

$$\begin{aligned} \begin{aligned} \mathcal {D}_{f} u_{\varepsilon ;t}&= \int _0^t e^{(t-s)\Delta }\partial _x \hat{\Pi }_\varepsilon \left( 2u_{\varepsilon ;s}\mathcal {D}_f u_{\varepsilon ;s}\partial _x \rho _{u_{\varepsilon ;s}} + u^2_{\varepsilon ;s}\partial _x \rho _{\mathcal {D}_f u_{\varepsilon ;s}} \right) \,\mathop {}\!\mathrm {d}s \\&\quad + \sum _{m\in N_\varepsilon } \int _0^t \ell _\varepsilon (m )e^{-4\pi ^2|m|^2(t-s)}\mathop {}\!\mathrm {d}f_{m;s}e_m. \end{aligned} \end{aligned}$$

(5.13)

Finally, for $\varepsilon \in (0,\varepsilon _0)$, there exists a $C(T,\,\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}},\,\Vert Z_\varepsilon \Vert _{C_T\mathcal {C}^{\alpha }})>0$ such that,

$$\begin{aligned} \Vert \mathcal {D}_f u_{\varepsilon }\Vert _{C_{T}\mathcal {C}^{\alpha }} \le C \Vert f\Vert _{C_T\mathbb {C}^{N_\varepsilon }}. \end{aligned}$$

(5.14)

Proof

Integration by parts implies that, for any $m\in \mathbb {Z}$ and $f\in C_T\mathbb {C}$ with $f_0=0$

$$\begin{aligned} \int _0^t e^{-4\pi ^2|m|^2(t-s)}\mathop {}\!\mathrm {d}f_s = f_t - 4\pi ^2|m|^2\int _0^t e^{-4\pi ^2|m|^2(t-s)}f_s\,\mathop {}\!\mathrm {d}s. \end{aligned}$$

It follows that $Z_{\varepsilon }$ is a bounded, linear function of $\hat{B}_\varepsilon$ with values in $C_T \Pi _\varepsilon L^2_0(\mathbb {T})$, and so is continuously Fréchet differentiable. Furthermore, for any $f \in \bar{C}_T\mathbb {C}^{N_\varepsilon }$ with $f_0=0$

$$\begin{aligned} \mathcal {D}_f Z_{\varepsilon ;t} = \sum _{m\in N_\varepsilon } \left( f_{m;t} - 4\pi ^2 |m|^2\int _0^t e^{-4\pi ^2|m|^2(t-s)}f_{m;s}\,\mathop {}\!\mathrm {d}s\right) \ell _\varepsilon (m)e_m, \end{aligned}$$

Regarding the approximate solution, $u_{\varepsilon ;t}$, the mappings $h \mapsto h^2$ and $h\mapsto \partial _x \rho _h$ are Fréchet differentiable on $\Pi _\varepsilon (L^2(\mathbb {T}))$, so the map

$$\begin{aligned} \Phi _T:(z,(f_m)_{m\in N_\varepsilon })\,&\mapsto -z + e^{\,\cdot \,\Delta }\zeta _\varepsilon + \int _0^{\,\cdot \,} e^{(\,\cdot \, -s)\Delta } \partial _x \hat{\Pi }_\varepsilon \left( z_{s}^2 \partial _x \rho _{z_{s}}\right) \,\mathop {}\!\mathrm {d}s\\&\quad +\sum _{m\in N_\varepsilon } \int _0^{\,\cdot \,}\ell _\varepsilon (m) e^{-4\pi ^2|m|^2(t-s)}\mathop {}\!\mathrm {d}f_{m;s} \,e_m, \end{aligned}$$

is Fréchet differentiable as a mapping $\Phi _T:(C_{T}\Pi _\varepsilon L^2(\mathbb {T}),\bar{C}_T\mathbb {C}^{N_\varepsilon }) \rightarrow C_T\Pi _\varepsilon L^2(\mathbb {T})$ and is such that $\Phi _T(u_\varepsilon , \hat{B}_\varepsilon ) =0$. Moreover, writing $\mathscr {D}$ for the Fréchet derivative,

$$\begin{aligned} (\mathscr {D}\Phi _{T_*})(u_\varepsilon ,\hat{B}_\varepsilon )(\,\cdot \,,0) :C_{T_*}\Pi _\varepsilon L^2(\mathbb {T}) \rightarrow C_{T_*}\Pi _\varepsilon L^2(\mathbb {T}) \end{aligned}$$

is a Banach space isomorphism for $T_*(\Vert u_\varepsilon \Vert _{C_T\mathcal {C}^\alpha })>0$ sufficiently small. Applying the implicit function theorem for Banach spaces, [1, Thm. 2.3], we obtain that $u_\varepsilon (\zeta ,\cdot )|_{[0,T_*]}$ is Fréchet differentiable in a neighbourhood of $\hat{B}_\varepsilon$. Patching together a sufficient (but finite) number of intervals of length $T_*$ to cover [0, T], we obtain the first claim.

To derive (5.13), recall that the Fréchet and Gateaux derivatives of a Fréchet differentiable map agree. Hence, for any $f\in \bar{C}_T\mathbb {C}^{N_\varepsilon }$, $\mathcal {D}_f u_{\varepsilon } = \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } \mathscr {S}^\varepsilon (\zeta ,\hat{B}_\varepsilon + \lambda f_\varepsilon )\big |_{\lambda =0}$ from which (5.13) follows now from the approximate equation, (5.8). We finally show (5.14). By the triangle inequality, the properties of $\hat{\Pi }_\varepsilon$, (A.13) and applying (A.10), for any $t\in [0,T]$,

$$\begin{aligned} \Vert \mathcal {D}_f u_{\varepsilon ;t}\Vert _{\mathcal {C}^{\alpha }} \lesssim \Vert u_{\varepsilon }\Vert ^2_{C_{\eta ;t}\mathcal {C}^{\alpha }} \int _0^t s^{-\frac{1}{2}-2\eta } \Vert \mathcal {D}_f u_{\varepsilon ;s}\Vert _{\mathcal {C}^{\alpha }}\,\mathop {}\!\mathrm {d}s + \sup _{s\in [0,t]}|f_s|_{\mathbb {C}^{N_\varepsilon }}. \end{aligned}$$

Therefore, by Grönwall’s inequality, there exists $C>0$ such that

$$\begin{aligned} \Vert \mathcal {D}_f u_{\varepsilon ;t}\Vert _{\mathcal {C}^{\alpha }} \lesssim \Vert f\Vert _{C_T \mathbb {C}^{N_\varepsilon }} \exp \big ( Ct^{\frac{1}{2}-2\eta }\Vert u_\varepsilon \Vert ^2_{C_{\eta ;t}\mathcal {C}^{\alpha }} \big ). \end{aligned}$$

Due to the global existence of $u_{\varepsilon }\in C_{\eta ;T}\mathcal {C}^{\alpha }$ (shown in Theorem 5.8 for $\varepsilon \in (0,\varepsilon _0)$ where $\varepsilon _0$ depends on $\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}, \Vert Z_\varepsilon \Vert _{C_T\mathcal {C}^{\alpha }}$), for a new constant $C(T,\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}},\Vert Z\Vert _{C_T\mathcal {C}^{\alpha }}) >0$,

$$\begin{aligned} \Vert \mathcal {D}_f u_{\varepsilon }\Vert _{C_{T}\mathcal {C}^{\alpha }} \le C \Vert f\Vert _{C_{ T}\mathbb {C}^{N_\varepsilon }}. \end{aligned}$$

$\square$

Regarding the derivative of $u_\varepsilon (\zeta )$ with respect to the initial condition, for $g \in \mathcal {C}^{\alpha _0}$, we set $g_\varepsilon := \hat{\Pi }_\varepsilon g$ and then for any $0\le s\le t \le T$ we let $J_{s,t}^\varepsilon g$ solve the equation

$$\begin{aligned} J^\varepsilon _{s,t}g = e^{(t-s)\Delta } g_\varepsilon + \int _s^t e^{(t-r)\Delta }\partial _x \hat{\Pi }_\varepsilon [2u_{\varepsilon ;r} \partial _x \rho _{u_{\varepsilon ;r}}J_{s,r}^\varepsilon g + u^2_{\varepsilon ;r}\partial _x \rho _{J_{s,r}^\varepsilon g}]\,\mathop {}\!\mathrm {d}r. \end{aligned}$$

(5.15)

We show below that for any $\zeta \in \mathcal {C}^{\alpha _0}(\mathbb {T})$ and $t\in [0,T]$, $J^\varepsilon _{0,t}g = D_{g}u_\varepsilon (\zeta )$, the directional derivative of $u_\varepsilon (\zeta )$ in $\zeta$. Note that $J^\varepsilon$ satisfies the flow property, that is for $0\le s\le t \le T$ one has $J^{\varepsilon }_{0,t} = J^\varepsilon _{s,t}J^\varepsilon _{0,s}$. In particular $J^\varepsilon _{s,s}=\mathrm {id}$.

Lemma 5.12

There exists an open neighbourhood $\mathcal {O}_{\zeta _\varepsilon } \subset \Pi _\varepsilon (L^2(\mathbb {T}))$ containing $\zeta _\varepsilon$ such that $u_\varepsilon (\,\cdot \,,\hat{B}_\varepsilon )$ is Fréchet differentiable as a mapping from $\mathcal {O}_{\zeta _\varepsilon }$ to $C_{T}\Pi _\varepsilon (L^2(\mathbb {T}))$. For any $g \in \mathcal {C}^{\alpha _0}$, the derivative is given by $D_{g} u_{\varepsilon ;t}(\zeta ) = J_{0,t}^\varepsilon g$. Furthermore, setting $\mathfrak {R}=\Vert Z\Vert _{C_T\mathcal {C}^{\alpha }}+\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}$, there exists $C(\mathfrak {R})>0$ and $T_*(\mathfrak {R})>0$ such that for all $t \in (0,T_*]$

$$\begin{aligned} \sup _{s \in [0,t]} s^\eta \Vert J_{0,s}^\varepsilon g\Vert _{\mathcal {C}^{\alpha }} \le C \Vert g\Vert _{\mathcal {C}^{\alpha _0}}. \end{aligned}$$

(5.16)

Proof

The same argument as in the proof of Lemma 5.11 shows that the map $\Pi _\varepsilon L^2_\mathfrak {m}(\mathbb {T})\ni \zeta _\varepsilon \mapsto u_{\varepsilon }(\zeta )\in C_T\Pi _\varepsilon (L^2_\mathfrak {m}(\mathbb {T}))$ is Fréchet differentiable in a neighbourhood of $\zeta _\varepsilon$. It is then readily checked that on $\mathcal {O}_{\zeta _\varepsilon }$, for any $g \in \mathcal {C}^{\alpha _0}$ the Fréchet derivative is equal to the map $t\mapsto J^\varepsilon _{0,t}g$.

To prove (5.16), observe that

$$\begin{aligned} \Vert J^\varepsilon _{0,s}g \Vert _{\mathcal {C}^{\alpha }} \lesssim s^{-\frac{\alpha -\alpha _0}{2}}\Vert g\Vert _{\mathcal {C}^{\alpha _0}} + \Vert u_{\varepsilon }\Vert ^2_{C_{\eta ;s}\mathcal {C}^{\alpha }}\int _0^s (s-r)^{-\frac{1}{2}}r^{-2\eta }\Vert J^\varepsilon _{0,r}g\Vert _{\mathcal {C}^{\alpha }}\,\mathop {}\!\mathrm {d}r. \end{aligned}$$

Therefore, for any $t\in (0,T]$,

$$\begin{aligned} \sup _{s\in [0,t]}s^\eta \Vert J^\varepsilon _{0,s}g \Vert _{\mathcal {C}^{\alpha }} \lesssim t^{\eta -\frac{\alpha -\alpha _0}{2}}\Vert g\Vert _{\mathcal {C}^{\alpha _0}} + t^{\frac{1}{2}-\eta }\Vert u_{\varepsilon }\Vert ^2_{C_{\eta ;t}\mathcal {C}^{\alpha }}\sup _{s\in [0,t]}s^\eta \Vert J^\varepsilon _{0,s}g \Vert _{\mathcal {C}^{\alpha }} . \end{aligned}$$

Since $\Vert u_\varepsilon -u\Vert _{C_{\eta ;T}\mathcal {C}^\alpha }\le 1$, by Theorem 3.4 there exists a $T_*(\mathfrak {R})\in (0,1)$ such that $\Vert u_{\varepsilon }\Vert _{C_{\eta ;T_*}\mathcal {C}^{\alpha }}\le 2$. Hence, for all $t\in (0,T_*]$,

$$\begin{aligned} \sup _{s \in [0,t]}s^\eta \Vert J^\varepsilon _{0,s}g \Vert _{\mathcal {C}^{\alpha }} \lesssim t^{\eta -\frac{\alpha -\alpha _0}{2}}\Vert g\Vert _{\mathcal {C}^{\alpha _0}} + t^{\frac{1}{2}-\eta }\sup _{s \in [0,t]}s^\eta \Vert J^\varepsilon _{0,s}g \Vert _{\mathcal {C}^{\alpha }}, \end{aligned}$$

so then choosing a sufficiently small time $t_1(\mathfrak {R}) \in (0,t]$,

$$\begin{aligned} \sup _{s \in [0,t_1]}s^\eta \Vert J^\varepsilon _{0,s}g \Vert _{\mathcal {C}^{\alpha }} \lesssim t_1^{\eta -\frac{\alpha -\alpha _0}{2}}\Vert g\Vert _{\mathcal {C}^{\alpha _0}}. \end{aligned}$$

Repeating this procedure, we find a constant $C:=C(\mathfrak {R})>0$ such that

$$\begin{aligned} \sup _{s \in [0,t]}s^\eta \Vert J^\varepsilon _{0,s}g \Vert _{\mathcal {C}^{\alpha }} \le C \Vert g\Vert _{\mathcal {C}^{\alpha _0}},\quad \forall \, t \in (0,T_*]. \end{aligned}$$

$\square$

We finally reintroduce probability and consider in the remainder of the section a finite dimensional approximation $B_{\varepsilon ;t}$ of the white noise defined by

$$\begin{aligned} B_{\varepsilon ;t} := \sum _{m\in N_\varepsilon } e_m\hat{B}_{t;m},\quad \hat{B}_{m;t} := \xi (\mathbbm {1}_{[0,t]}\times e_m), \quad \text { for }m \in N_\varepsilon . \end{aligned}$$

Note that $(\hat{B}_m)_{m\in N_\varepsilon }$ is a family of complex valued Brownian motions which satisfy the reality condition (5.12). Our approximation of the SHE corresponding to (5.11) is then

$$\begin{aligned} v_{\varepsilon ;s,t}:=\hat{\Pi }_\varepsilon v_{s,t} =\sum _{m\in N_\varepsilon } \int _s^t \ell _\varepsilon (m) e^{-4\pi ^2|m|^2(t-r)}\mathop {}\!\mathrm {d}\hat{B}_{m;r} \,e_m, \end{aligned}$$

(5.17)

By Remark 5.10, since $\hat{B}_\varepsilon \in \bar{C}_T\mathbb {C}^{N_\varepsilon }$, $(s,t)\mapsto v_{\varepsilon ;s,t}$ is well defined pathwise. Furthermore, by Property ii), there exists a $\mathbb {P}$-null set $\mathcal {N}\subset \Omega$ such that, fixing any realisation $\xi (\omega )$ for $\omega \in \Omega \setminus \mathcal {N}$ gives realisations of $v_{\varepsilon }(\omega ),\,v(\omega )$ and for which $v_{\varepsilon ;0,\,\cdot \,}(\omega )\rightarrow v_{0,\,\cdot \,}(\omega )$ in $\mathcal {C}^\kappa _T\mathcal {C}^{\alpha -2\kappa }$ for every $\kappa \in [0,1/2)$. In the rest of the subsection, we will let $v_\varepsilon$ denote the random path $t\mapsto v_{\varepsilon ;0,t}$.

For each $\varepsilon >0$, the Cameron–Martin space of $\hat{B}_\varepsilon$ is

$$\begin{aligned} \mathcal {C}\mathcal {M}^\varepsilon = \bar{W}^{1,2}_0\left( [0,T];\mathbb {C}^{N_\varepsilon }\right) :=&\bigg \{ f \in L^2([0,T];\,\mathbb {C}^{N_\varepsilon })\,:\, \partial _t f \in L^2\left( [0,T];\,\mathbb {C}^{N_\varepsilon }\right) , \,\\&f_{-m} = \bar{f}_m,\, f|_{t=0}=0 \bigg \}. \end{aligned}$$

By the Sobolev embedding, $W^{1,2}(\mathbb {R})\hookrightarrow \mathcal {C}^{1/2}(\mathbb {R})$, we see $\mathcal {C}\mathcal {M}^\varepsilon \subset \bar{C}_T \mathbb {C}^{N_\varepsilon }$. Therefore, Lemma 5.11 applies with $f \in \mathcal {C}\mathcal {M}^\varepsilon$. We also choose a smooth, compactly supported, cut-off function $\Theta :\mathbb {R}_+\rightarrow [0,1]$ such that $\Theta (z) = 1$ for $z<\frac{1}{2}$ and $\Theta (z)=0$ for $z\ge 1$.

We introduce the notion of right sided derivatives, $\mathcal {D}^+$, of $\Vert Z\Vert _{C_T\mathcal {C}^\alpha }$, which, for $f \in \bar{C}_T\mathcal {C}^{\alpha }$, is defined by

$$\begin{aligned} \mathcal {D}^+_{f} \Vert Z\Vert _{C_T\mathcal {C}^{\alpha }}:= \lim _{\lambda \searrow 0} \frac{\Vert Z+\lambda f\Vert _{C_T\mathcal {C}^{\alpha }} -\Vert Z\Vert _{C_T\mathcal {C}^{\alpha }}}{\lambda }. \end{aligned}$$

Theorem 5.13

(Bismut–Elworthy–Li Formula) Let $t>0$, $\zeta \in \mathcal {C}^{\alpha _0}_\mathfrak {m}(\mathbb {T})$, $\Phi \in \mathcal {C}^1_b(\mathcal {C}^{\alpha _0}(\mathbb {T}))$ and $f^\varepsilon$ be a $\mathcal {C}\mathcal {M}^\varepsilon$ valued process taking for which $\partial _t f^\varepsilon$ is adapted and such that $\Vert \partial _t f^\varepsilon \Vert _{L^2([0,t]; \mathbb {C}^{N_\varepsilon })}\le C$, $\mathbb {P}$-a.s. for a deterministic constant $C:=C(t)$. Then we have the identity

$$\begin{aligned}&\mathbb {E}\big [ \mathcal {D}_{f^\varepsilon } [\Phi (u_{\varepsilon ;t}(\zeta ,\hat{B}_\varepsilon ))]\Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})\big ]\nonumber \\&\quad = \mathbb {E}\Big [ \Phi (u_{\varepsilon ;t}(\zeta ,\hat{B}_\varepsilon )) \Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})\int _0^t \partial _t f^\varepsilon _s \cdot \,\mathop {}\!\mathrm {d}\hat{B}_{\varepsilon ;s} \Big ]\nonumber \\&\qquad - \mathbb {E}\Big [\Phi (u_{\varepsilon ;t}(\zeta ,\hat{B}_\varepsilon ))\Theta '(\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})\mathcal {D}^+_{f^\varepsilon } \Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }} \Big ], \end{aligned}$$

(5.18)

where we recall that $f^\varepsilon$ is identified with a function in $C_t C^\infty _0$ as in Remark 5.10.

Proof

For any $f^\varepsilon \in \mathcal {C}\mathcal {M}^\varepsilon$ and $\lambda >0$, define the shift operator $\mathcal {T}^\lambda _{f^\varepsilon } \hat{B}_\varepsilon := \hat{B}_\varepsilon +\lambda f^\varepsilon$. Define further

$$\begin{aligned} u^{\lambda ;f}_{\varepsilon }(\zeta )&:= \mathscr {S}^\varepsilon \left( \zeta , \mathcal {T}^\lambda _{f^\varepsilon } \hat{B}^\varepsilon \right) ,\\ \mathcal {T}^\lambda _{f^\varepsilon } v_{\varepsilon ;0,t}&:=\sum _{m\in N_\varepsilon } \int _0^t \ell _\varepsilon (m) e^{-4\pi ^2|m|^2(t-r)}(\mathop {}\!\mathrm {d}\hat{B}^m_r +\lambda \partial _t f^\varepsilon _{m;r} \,\mathop {}\!\mathrm {d}r)\,e_m. \end{aligned}$$

We now look for a family of measures $\mathbb {P}_\lambda$ such that the law of $\mathcal {T}^\lambda _{f^\varepsilon }\hat{B}^\varepsilon$ under $\mathbb {P}_\lambda$ is equal to the law of $\hat{B}^\varepsilon$ under $\mathbb {P}$ for each $\lambda$. The assumptions on $f^\varepsilon$ imply that Novikov’s condition [30, Ch. 8, Prop. 1.15] is satisfied by $\lambda \partial _t f^\varepsilon$. Hence, by Girsanov’s theorem [30, Ch. 4, Thm. 1.4], the desired measure $\mathbb {P}_\lambda$ is given by

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}\mathbb {P}_\lambda }{\mathop {}\!\mathrm {d}\mathbb {P}} := A^\lambda _t, \quad A_t^\lambda := \exp \left( -\lambda \int _0^t \partial _t f^\varepsilon _s \,\cdot \mathop {}\!\mathrm {d}\hat{B}^\varepsilon _s -\frac{\lambda ^2}{2}\int _0^t |\partial _tf^\varepsilon _s|^2\,\mathop {}\!\mathrm {d}s\right) . \end{aligned}$$

As a consequence, and using $\mathbb {E}_{\mathbb {P}_\lambda }\left[ \,\cdot \,\right] = \mathbb {E}\left[ \,\cdot \, A^\lambda _t\right]$, we have

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda }\mathbb {E}\left[ \Phi (u^{\lambda ;f^\varepsilon }_{\varepsilon ;t}(\zeta ))\Theta (\Vert \mathcal {T}^\lambda _{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\lambda }_t \right] \bigg |_{\lambda =0+}= 0. \end{aligned}$$

We now show that we may exchange differentiation and integration in the above expression, after which we will derive the BEL formula, (5.18).

Let us define the difference operator $\Delta _\lambda F := F(\lambda )-F(0)$ so that, for example,

$$\begin{aligned} \Delta _\lambda \Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(\zeta )) = \Phi (u^{\lambda ;f^\varepsilon }_{\varepsilon ;t}(\zeta ))- \Phi (u_{\varepsilon ;t}(\zeta )). \end{aligned}$$

We demonstrate that the family $\{\frac{1}{\lambda }(\Delta _{\lambda }(\Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(x))\Theta (\Vert \mathcal {T}^{\,\cdot \,}_{f^\varepsilon } v_{\varepsilon ;0,\,\cdot \,}\Vert _{C_T\mathcal {C}^{\alpha }})A^{\,\cdot \,}_t))\}_{\lambda \in (0,1]}$ is $\mathbb {P}$-uniformly integrable. For any $\lambda \in (0,1]$ we observe that,

$$\begin{aligned} \Delta _{\lambda }(\Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(\zeta ))\Theta (\Vert \mathcal {T}^{\,\cdot \,}_{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\,\cdot \,}_t)&= \Delta _{\lambda }\Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(\zeta ))\Theta (\Vert \mathcal {T}^{\lambda }_{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\lambda }_t\\&\quad +\Phi (u_{\varepsilon ;t}(\zeta ))\Delta _{\lambda }\Theta (\Vert \mathcal {T}^{\,\cdot \,}_{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\lambda }_t \\&\quad +\Phi (u_{\varepsilon ;t}(\zeta ))\Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})\Delta _{\lambda } A^{\,\cdot \,}_t \end{aligned}$$

For the first term, by the assumption that $\Phi \in \mathcal {C}^1_b(\mathcal {C}^{\alpha _0}(\mathbb {T}))$,

$$\begin{aligned} |\Delta _{\lambda }(\Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(\zeta ))|&\le \Vert \Phi \Vert _{\mathcal {C}^{1}_b} \Vert u^{\lambda ;f^\varepsilon }_{\varepsilon ;t}(\zeta ) - u_{\varepsilon ;t}(\zeta )\Vert _{\mathcal {C}^{\alpha _0}}\\&\le \Vert \Phi \Vert _{\mathcal {C}^{1}_b} \int _0^\lambda \left\| \mathcal {D}_{f}u^{\tilde{\lambda };f}_{\varepsilon ;t}(\zeta )\right\| _{\mathcal {C}^{\alpha }}\mathop {}\!\mathrm {d}\tilde{\lambda }. \end{aligned}$$

If $\Vert \mathcal {T}^{{\lambda }}_{f^\varepsilon }v_\varepsilon \Vert _{C_t\mathcal {C}^{\alpha }} > 1$, then the first term vanishes, so suppose that $\Vert \mathcal {T}^{{\lambda }}_{f^\varepsilon }v_\varepsilon \Vert _{C_t\mathcal {C}^{\alpha }}\le 1$. Hence, due to the bound on $f^\varepsilon$, there exists a deterministic $C>0$ such that $\Vert \mathcal {T}^{\tilde{\lambda }}_{f^\varepsilon }v_\varepsilon \Vert _{C_t\mathcal {C}^{\alpha }}\le C$ for all $\tilde{\lambda }\in [0,\lambda ]$. Therefore, applying (5.14) uniformly for $\tilde{\lambda }\in (0,\lambda ]$, we have the bound

$$\begin{aligned} |\Delta _{\lambda }\Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(\zeta ))\Theta (\Vert \mathcal {T}^{\lambda }_{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\lambda }_t| \,\lesssim _{f,\Theta ,t} \lambda \Vert \Phi \Vert _{\mathcal {C}^1_b} A^\lambda _t, \end{aligned}$$

where the implicit constant is independent of the realisation of v and $\lambda \in (0,1]$. For the second term, arguing similarly and using our assumptions on $\Theta$ we obtain the similar bound,

$$\begin{aligned} |\Phi (u_{\varepsilon ;t}(\zeta ))\Delta _{\lambda }\Theta (\Vert \mathcal {T}^{\,\cdot \,}_{f^\varepsilon } v_{\varepsilon }\Vert _{C_T\mathcal {C}^{\alpha }})A^{\lambda }_t|\,\lesssim _{f,\Theta ,t} \lambda \Vert \Phi \Vert _{\infty } A^\lambda _t. \end{aligned}$$

Finally, for the third term,

$$\begin{aligned} |\Phi (u_{\varepsilon ;t}(\zeta ))\Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})\Delta _{\lambda } A^{\,\cdot \,}_t| \lesssim _{\Theta } \Vert \Phi \Vert _{\infty } \int _0^\lambda \left| \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } A^{\tilde{\lambda }}_t\right| \mathop {}\!\mathrm {d}\tilde{\lambda }. \end{aligned}$$

Therefore there exists a constant $C(f,\Theta ,t,\Vert \Phi \Vert _{\mathcal {C}^1_b})>0$ such that for all $\lambda \in (0,1]$ and $p\ge 1$

$$\begin{aligned}&\mathbb {E}\left[ \left| \frac{ \Delta _{\lambda }(\Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(\zeta ))\Theta (\Vert \mathcal {T}^{\,\cdot \,}_{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\,\cdot \,}_t)}{\lambda }\right| ^p\right] \\&\quad \le C \left( \sup _{\lambda \in (0,1]}\mathbb {E}[(A^\lambda _t)^p]+ \sup _{\lambda \in (0,1]}\mathbb {E}\left[ \left| \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } A^{\lambda }_t\right| ^p\right] \right) , \end{aligned}$$

where we applied Jensen’s inequality to obtain $(\int _0^\lambda |\frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } A^{\tilde{\lambda }}_t|\mathop {}\!\mathrm {d}\tilde{\lambda })^p{\le }\lambda ^{p-1}\int _0^\lambda |\frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } A^{\tilde{\lambda }}_t|^p\mathop {}\!\mathrm {d}\tilde{\lambda }$. For the first term we have $(A^\lambda _t)^p = A^{p\lambda }_t \exp \left( \frac{p^2-p}{2}\lambda ^2\int _0^t |\partial _t f^\varepsilon _s|^2\mathop {}\!\mathrm {d}s\right)$ where $A^{p\lambda }$ is also a strictly positive martingale with expectation 1 and $\exp \left( \frac{p^2-p}{2}\lambda ^2\int _0^t |\partial _t f^\varepsilon _s|^2\mathop {}\!\mathrm {d}s\right)$ is uniformly bounded in $L^\infty (\Omega ;\mathbb {R})$ across $\lambda \in (0,1]$ due to the assumptions on $f^\varepsilon$. Regarding the second term, using the explicit form,

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } A^{\lambda }_t= -A^{\lambda }_t \left( \int _0^t \partial _t f^\varepsilon _s\cdot \mathop {}\!\mathrm {d}\hat{B}_{\varepsilon ;s} + \lambda \int _0^t|\partial _t f^\varepsilon _s|^2\mathop {}\!\mathrm {d}s \right) \end{aligned}$$

(5.19)

and Cauchy–Schwarz, we see that

$$\begin{aligned} \mathbb {E}\left[ \left| \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } A^{\tilde{\lambda }}_t\right| ^p\right] ^2 \le \mathbb {E}\left[ (A^{\tilde{\lambda }}_t)^{2p}\right] \mathbb {E}\left[ \left( \int _0^t \partial _t f^\varepsilon _s\cdot \mathop {}\!\mathrm {d}\hat{B}_s^\varepsilon + \tilde{\lambda } \int _0^t|\partial _t f^\varepsilon _s|^2\mathop {}\!\mathrm {d}s \right) ^{2p}\right] . \end{aligned}$$

The first factor is uniformly bounded for all $\tilde{\lambda } \in (0,1]$ by the same arguments as we applied for $(A^\lambda _t)^p$. The second factor can be controlled using the Burkholder–Davis–Gundy inequality and again our assumption of uniform boundedness on $\Vert \partial _tf^\varepsilon \Vert _{L^2([0,t])}$. Hence

$$\begin{aligned} \sup _{\lambda \in (0,1]}\mathbb {E}\left[ \,\left| \frac{ \Delta _{\lambda }(\Phi (u^{\,\cdot \,;f^\varepsilon }_{\varepsilon ;t}(x))\Theta (\Vert \mathcal {T}^{\,\cdot \,}_{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\,\cdot \,}_t)}{\lambda }\right| ^p\,\right] <\infty , \end{aligned}$$

and we have uniform integrability. Therefore, by Vitali’s convergence theorem, we may exchange differentiation and expectation to give that

$$\begin{aligned} \mathbb {E}\left[ \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda } \left( \Phi (u^{\lambda ;f^\varepsilon }_{\varepsilon ;t}(\zeta ))\Theta (\Vert \mathcal {T}^\lambda _{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})A^{\lambda }_t\right) \bigg |_{\lambda =0+} \right] = 0. \end{aligned}$$

In order to conclude the proof it only remains to show that the identities

$$\begin{aligned} \mathcal {D}^+_{f^\varepsilon }\Phi \left( u_{\varepsilon ;t}(\zeta )\right)&= \mathcal {D}_{f^\varepsilon }\Phi \left( u_{\varepsilon ;t}(\zeta )\right) ,\\ \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda }A^{\lambda }_t \bigg |_{\lambda =0+}&= -\int _0^t \partial _t f^\varepsilon _s\cdot \mathop {}\!\mathrm {d}\hat{B}_{\varepsilon ;s},\\ \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}\lambda }\Theta \left( \Vert \mathcal {T}^\lambda _{f^\varepsilon } v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }}\right) \bigg |_{\lambda =0+}A^{0 }_t&=\Theta '(\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})\mathcal {D}^+_{f^\varepsilon } \Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }}. \end{aligned}$$

The first follows directly from Lemma 5.11, since one sided derivatives and full derivatives agree for any Fréchet differentiable map. The second identity follows after setting $\lambda =0$ in (5.19) and observing that $A^0_t\equiv 1$. The final identity follows in the same way from our assumptions on $\Theta$ and the chain rule. $\square$

We can apply the identity (5.18) to obtain an initial bound on the difference of the semi-group acting on elements of $\mathcal {C}^1_b(\mathcal {C}^{\alpha _0})$, from two initial points. The following result and its proof is close to that of [34, Prop. 5.8].

Proposition 5.14

Let $\zeta ,\,\tilde{\zeta } \in \mathcal {C}_\mathfrak {m}^{\alpha _0}$ with $\tilde{\zeta } \in B_1(\zeta )$, set $\mathfrak {R}:= 1+2\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}}$ and let $T_*(\mathfrak {R})>0$ be defined according to (3.7). Then there exists a constant $C:=C(\mathfrak {R},\alpha ,\alpha _0,\eta )>0$ and exponent $\theta :=\theta (\eta )>0$ such that for any $\Phi \in \mathcal {C}^1_{b}(\mathcal {C}^{\alpha _0}_\mathfrak {m})$ and $t \in (0,T_*]$,

$$\begin{aligned} |P_t\Phi (\zeta )-P_t\Phi (\tilde{\zeta })| \le C\frac{1}{t^\theta } \Vert \Phi \Vert _{L^\infty } \Vert \zeta -\tilde{\zeta }\Vert _{\mathcal {C}^{\alpha _0}} + 2\Vert \Phi \Vert _{L^\infty } \mathbb {P}(\Vert v\Vert _{C_t\mathcal {C}^{\alpha }}>1). \end{aligned}$$

(5.20)

Proof

First, defining the semi-group for the approximate equation and applying the triangle inequality, for every $\Phi \in \mathcal {C}^1_b(\mathcal {C}^{\alpha _0})$ one obtains

$$\begin{aligned} |P^\varepsilon _t\Phi (\zeta )-P^\varepsilon _t\Phi (\tilde{\zeta })|&\le \left| \mathbb {E}\left[ (\Phi (u_{\varepsilon ;t}(\zeta )) - \Phi (u_{\varepsilon ;t}(\tilde{\zeta })))\Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha } } ) \right] \right| \nonumber \\&\quad + \left| \mathbb {E}\left[ (\Phi (u_{\varepsilon ;t}(\zeta ))-\Phi (u_{\varepsilon ;t}(\zeta )))\left( 1-\Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})\right) \right] \right| \nonumber \\&=: I_1(t) + I_2(t). \end{aligned}$$

(5.21)

For the second term,

$$\begin{aligned} I_2(t) \le 2 \Vert \Phi \Vert _{L^\infty }\mathbb {P}\left( \Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }}>1 \right) , \end{aligned}$$

(5.22)

and using the fact that $v_\varepsilon \rightarrow v$ in $C_t\mathcal {C}^{\alpha }$ in probability we obtain the second term on the right hand side of (5.20). Regarding the first term, we use the fundamental theorem of calculus along with Fubini to write,

$$\begin{aligned} I_1(t)= \left| \int _0^1 \mathbb {E}\left[ D_{\zeta -\tilde{\zeta }} [\Phi (u_t(\zeta +\lambda (\zeta -\tilde{\zeta }))] \Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }} ) \right] \mathop {}\!\mathrm {d}\lambda \,\right| . \end{aligned}$$

We now note that for any $f^\varepsilon \in \mathcal {C}\mathcal {M}^\varepsilon$, it follows from the mild forms of $\mathcal {D}_{f^\varepsilon }u_{\varepsilon ;t}$ and $J^\varepsilon _{s,t}\partial _t f^\varepsilon _{m;s}$ given in (5.13) and (5.15) respectively, that,

$$\begin{aligned} \mathcal {D}_{f^\varepsilon } u_{\varepsilon ;t}(\zeta ) = \sum _{m\in N_\varepsilon }\int _0^t J^\varepsilon _{s,t}\partial _t f^\varepsilon _{m;s} e_m \,\mathop {}\!\mathrm {d}s. \end{aligned}$$

Denote $z_\lambda := \zeta -\lambda (\zeta -\tilde{\zeta })\in \mathcal {C}^{\alpha _0}$ and let $f^\varepsilon \in \mathcal {C}\mathcal {M}^\varepsilon$ such that $\partial _t f^{\varepsilon }_{m;s} = \frac{1}{t}\langle J^\varepsilon _{0,s}(\zeta -\tilde{\zeta }),e_m\rangle$ for every $m\in N_\varepsilon$, where $J^{\varepsilon }_{0,s}(\zeta -\tilde{\zeta }) = D_{\zeta -\tilde{\zeta }} u_{\varepsilon ;s}(z_\lambda )$ and for notational ease, we have suppressed the dependence on $z_\lambda$ in $J^{\varepsilon }_{0,s}(\zeta -\tilde{\zeta })$. Then

$$\begin{aligned} \mathcal {D}_{f^\varepsilon } u_{\varepsilon ;t}(z_\lambda ) = J_{0,t}^\varepsilon (\zeta -\tilde{\zeta }) = D_{\zeta -\tilde{\zeta }}u_{\varepsilon ;t}(z_\lambda ). \end{aligned}$$

(5.23)

Furthermore, for $f^\varepsilon$ defined in this way, $\partial _t f^\varepsilon \in L^\infty (\Omega ;L^2([0,t];\mathbb {C}^{N_\varepsilon }))$. Note also that $\Vert z_\lambda \Vert _{\mathcal {C}^{\alpha _0}}\le \mathfrak {R}$ for all $\lambda \in (0,1)$, so the local bounds of Lemmas 5.11 and 5.12 both hold uniformly in $z_\lambda$. Using (5.23) and (5.18) we obtain the bound

$$\begin{aligned}&\left| \mathbb {E}\left[ D_{\zeta -\tilde{\zeta }}[ \Phi (u_{\varepsilon ;t}(z_\lambda ))] \Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }}) \right] \right| \\&\quad \le \frac{1}{t}\Vert \Phi \Vert _{L^\infty } \Big ( \mathbb {E}\Big [ \,\Big | \int _0^t\langle J^{\varepsilon }_{0,s}(\zeta -\tilde{\zeta }),\mathop {}\!\mathrm {d}W_{\varepsilon ;s}\rangle \,\Theta (\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha } } ) \Big |\,\,\Big ]\\&\qquad + \mathbb {E}\left[ \,\left| \, \Theta '(\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha } } )\mathcal {D}_{ f^\varepsilon }^+\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }} \,\right| \,\right] \Big ). \end{aligned}$$

For the first term, one can apply Cauchy–Schwarz, Itô’s isometry and the results of Lemma 5.12 to obtain

$$\begin{aligned} \mathbb {E}\left[ \,\left| \int _0^t\langle J^{\varepsilon }_{0,s}(\zeta -\tilde{\zeta }),\mathop {}\!\mathrm {d}W_{\varepsilon ;s}\rangle \,\Theta (\Vert v_{\varepsilon ;0,\,\cdot \,}\Vert _{C_t\mathcal {C}^{\alpha } } ) \right| \right]&\le \mathbb {E}\left[ \int _0^t \Vert J^{\varepsilon }_{0,s}(\zeta -\tilde{\zeta })\Vert ^2_{L^2}\mathop {}\!\mathrm {d}s \right] ^{1/2}\nonumber \\&\lesssim _{\mathfrak {R},\alpha ,\alpha _0,\eta } t^{\frac{1}{2}-\eta }\Vert \zeta -\tilde{\zeta }\Vert _{\mathcal {C}^{\alpha _0}}.\nonumber \\ \end{aligned}$$

(5.24)

For the second term, using the explicit form of the directional derivative and Lemma 5.12 again applied to our choice of $f^\varepsilon$, we have

$$\begin{aligned} \mathbb {E}\left[ \,\left| \, \Theta '(\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha } } )\mathcal {D}_{ f^\varepsilon }^+\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }} \,\right| \,\right]&\le t^{-\eta }\mathbb {E}\left[ \sup _{s \in [0,t]}s^\eta \Vert J^\varepsilon _{0,s}(\zeta -\tilde{\zeta })\Vert _{\mathcal {C}^{\alpha }}|\Theta '(\Vert v_{\varepsilon }\Vert _{C_t\mathcal {C}^{\alpha }})| \right] \nonumber \\&\le C t^{-\eta }\Vert \zeta -\tilde{\zeta }\Vert _{\mathcal {C}^{\alpha _0}}, \end{aligned}$$

(5.25)

where in the last line we also used that $\Theta '$ is bounded.

Combining (5.22), (5.24) and (5.25) yields the stated inequality, (5.20), for the approximate semi-group $P^\varepsilon$. Taking the limit $\varepsilon \rightarrow 0$ and applying the dominated convergence theorem concludes the proof. $\square$

We conclude this section by obtaining a local Hölder bound for the dual semi-group. It follows from this bound that, for all $t>0$, $P_t :\mathcal {B}_b(\mathcal {C}_\mathfrak {m}^{\alpha _0})\rightarrow \mathcal {C}_b(\mathcal {C}_\mathfrak {m}^{\alpha _0})$, which concludes our proof of the strong Feller property for $(P_t)_{t>0}$.

Theorem 5.15

Let $\zeta ,\,\tilde{\zeta } \in \mathcal {C}_\mathfrak {m}^{\alpha _0}$ with $\tilde{\zeta } \in B_1(\zeta )$. Then there exists $C>0$, $\theta \in (0,1)$ and $\sigma >0$ such that, for every $t \ge 1$,

$$\begin{aligned} \Vert P^*_{t}\delta _\zeta - P^*_{t}\delta _{\tilde{\zeta }}\Vert _{\mathrm {TV}} \,\,\le C(1+\Vert \zeta \Vert _{\mathcal {C}^{\alpha _0}})^\sigma \Vert \zeta -\tilde{\zeta }\Vert ^\theta _{\mathcal {C}^{\alpha _0}}. \end{aligned}$$

Sketch of Proof

It follows from [5, Lem. 7.1.5] that (5.20) is equivalent to the statement,

$$\begin{aligned} \Vert P^*_{t}\delta _\zeta - P^*_{t}\delta _{\tilde{\zeta }}\Vert _{\mathrm {TV}} \,\,\le C\frac{1}{t^\theta }\Vert \zeta -\tilde{\zeta }\Vert _{\alpha _0} + 2\mathbb {P}(\Vert v\Vert _{C_t\mathcal {C}^{\alpha }}>1). \end{aligned}$$

From this bound, one may follow exactly the steps of [34, Thm. 5.10]. The key idea is to use the Gaussian nature of v to obtain the control $\mathbb {P}(\Vert v\Vert _{C_t\mathcal {C}^{\alpha }}> 1) \lesssim t^{\theta _2}$ for some $\theta _2\in (0,1)$. Then we use the explicit form of $T_*(\mathfrak {R})$ defined in (3.7) and monotonicity of the map $t\mapsto \Vert P^*_{t}\delta _\zeta - P^*_{t}\delta _{\tilde{\zeta }}\Vert _{\mathrm {TV}}$ to obtain the final result. $\square$

5.3 Full support

We demonstrate that $u_T(\zeta )$, and thus any invariant measure $\nu _\zeta$ for $(P_t)_{t>0}$ as in Theorem 5.6, has full support in $\mathcal {C}^{1/2-\delta }_{\bar{\zeta }}(\mathbb {T})$ for any $\delta \in (0,1/2)$. In this subsection we are not concerned with the behaviour of the solution near zero, so until the start of Sect. 5.4 we just consider $\alpha \in (0,1/2)$ and $\mathfrak {m}\in \mathbb {R}$.

Let $L^2_0(\mathbb {R}_+\times \mathbb {T})$ be the space of square integrable functions on $\mathbb {R}_+\times \mathbb {T}$ such that for any $t\ge 0$, $\bar{f}_t=0$. Then for any $T\ge 0$ we define the Cameron–Martin space of $v :=v_{0,\,\cdot \,}$,

$$\begin{aligned} \mathscr {H}_T := \left\{ h :[0,T] \times \mathbb {T}\rightarrow \mathbb {R}\,:\, h_t =\int _0^t e^{(t-s)\Delta }f_s\,\mathop {}\!\mathrm {d}s,\, f \in L^2_0(\mathbb {R}_+ \times \mathbb {T}) \right\} , \end{aligned}$$

Note that by Theorems A.2 and A.6, $\mathscr {H}_T$ is continuously and densely embedded in $\{h\in C_T\mathcal {C}^{\alpha }_0\,:\,h(0)=0\}$. The following is a direct consequence of the Cameron–Martin theorem.

Lemma 5.16

(Theorem 3.6.1 of [3]) Let $T>0$ and $\mathcal {L}(v)= (v)\# \mathbb {P}\in \mathcal {P}(C_T\mathcal {C}^\alpha )$ be the law of v. Then $\mathrm {supp}(\mathcal {L}(v)) =\overline{\mathscr {H}_T}^{\Vert \,\cdot \,\Vert _{C_T\mathcal {C}^{\alpha }}}$.

In the following theorem, we treat $\mathcal {L}(u_T(\zeta ))$ as a probability measure on $\mathcal {C}^\alpha _\mathfrak {m}(\mathbb {T})$.

Theorem 5.17

Let $T>0$, $\zeta \in \mathcal {C}^{\alpha }_{\mathfrak {m}}(\mathbb {T})$. Then

$$\begin{aligned} \mathrm {supp}(\mathcal {L}(u_T(\zeta )))= \mathcal {C}^{\alpha }_{\mathfrak {m}}(\mathbb {T}). \end{aligned}$$

Proof

We first show

$$\begin{aligned} \overline{\left\{ \mathscr {S}_T(\zeta ,h)\,:\,h\in \mathscr {H}_T \right\} }^{\Vert \,\cdot \,\Vert _{\mathcal {C}^{\alpha }}}\subseteq \mathrm {supp}(\mathcal {L}(u_T(\zeta ))) . \end{aligned}$$

(5.26)

Recall that the map $\mathscr {S}_T(\zeta ,\,\cdot \,):C_T\mathcal {C}^{\alpha }_0(\mathbb {T})\rightarrow \mathcal {C}^{\alpha }_\mathfrak {m}$ is continuous and $\mathscr {H}_T \subset C_T\mathcal {C}^{\alpha }_0(\mathbb {T})$. Consider now $h\in \mathscr {H}_T$. Then for any $\varepsilon >0$, there exists a $\delta >0$ such that

$$\begin{aligned} \Vert v-h\Vert _{C_T\mathcal {C}^{\alpha }_0}<\delta \,\Rightarrow \,\Vert u_T-\mathscr {S}_T(\zeta ,h)\Vert _{\mathcal {C}^{\alpha }}<\varepsilon , \end{aligned}$$

and therefore

$$\begin{aligned} \mathbb {P}(\Vert u_T-\mathscr {S}_T(\zeta ,h)\Vert _{\mathcal {C}^{\alpha }}<\varepsilon ) \ge \mathbb {P}( \Vert v-h\Vert _{C_T\mathcal {C}^{\alpha }_0} <\delta ) >0, \end{aligned}$$

where the last inequality follows from Lemma 5.16 and this shows (5.26). It now suffices to show that,

$$\begin{aligned} \overline{\left\{ \mathscr {S}_T(\zeta ,h)\,:\,h\in \mathscr {H}_T \right\} }^{\Vert \,\cdot \,\Vert _{\mathcal {C}^{\alpha }}}= \mathcal {C}_\mathfrak {m}^{\alpha }(\mathbb {T}). \end{aligned}$$

Let $y\in C_{\mathfrak {m}}^\infty (\mathbb {T})$. Then for $t\in [0,T]$ define

$$\begin{aligned} u^y_t := e^{t\Delta }\zeta + \frac{t}{T}(y-e^{T\Delta }\zeta ), \end{aligned}$$

(5.27)

and

$$\begin{aligned} h^y_t := -\int _0^t e^{(t-s)\Delta } \partial _x((u^y_s)^2\partial _x \rho _{u^y_s}) \mathop {}\!\mathrm {d}s + \frac{t}{T}(y-e^{T\Delta }\zeta ). \end{aligned}$$

Since $\zeta \in \mathcal {C}^{\alpha }_\mathfrak {m}$, we have $u^y \in C_T\mathcal {C}_\mathfrak {m}^{\alpha }$ and it also follows that $h^y \in C_T\mathcal {C}_0^{\alpha }$ with $h^y(0)=0$. Furthermore, by construction,

$$\begin{aligned} \mathscr {S}_T(\zeta ,h^y)= u_T^y=y. \end{aligned}$$

(5.28)

Approximating $h^y$ by functions in $C^\infty _0([0,T]\times \mathbb {T})\cap \mathscr {H}_T$ and using the density of $C^\infty (\mathbb {T})$ in $\mathcal {C}^{\alpha }(\mathbb {T})$ concludes the proof. $\square$

5.4 Exponential mixing

In Theorem 5.18 and Corollary 5.19 below, we keep $(\alpha _0,\,\alpha )$ satisfying (5.6), the more restrictive parameter set from the start of Sect. 5.2.

Theorem 5.18

There exists $\lambda \in (0,1)$ such that for every $\zeta ,\,\tilde{\zeta } \in \mathcal {C}_\mathfrak {m}^{\alpha _0}$ and $t\ge 1$,

$$\begin{aligned} \Vert P^*_t\delta _\zeta -P^*_t \delta _{\tilde{\zeta }}\Vert _{\mathrm {TV}} \,\,\le \,\lambda . \end{aligned}$$

(5.29)

Sketch of Proof

It suffices to essentially repeat the proof of [34, Thm. 6.5]. Given an arbitrary fixed time interval, the idea is to divide the interval into three sub-intervals and repeatedly evolve the solutions, $u(\zeta ),\,u(\tilde{\zeta })$, for the duration of each sub-interval into an open bounded set, couple the semi-groups started from these points and successively apply the results of Theorems 5.3, 5.15 and 5.17. $\square$

Corollary 5.19

For any $\mathfrak {m}\in \mathbb {R}$, there exists a unique measure $\nu \in \mathcal {P}( \mathcal {C}^{\alpha _0}_{\mathfrak {m}})$ which is invariant for the semi-group $(P_t)_{t\ge 0}$. Furthermore, for any $\delta \in (0,1/2)$, $\nu (\mathcal {C}^{1/2-\delta }_\mathfrak {m})=1$ and $\nu$ has full support in $\mathcal {C}^{1/2-\delta }_\mathfrak {m}$. Finally, there exists $\lambda \in (0,1)$ such that for every $\mu \in \mathcal {P}(\mathcal {C}^{\alpha _0}_\mathfrak {m})$ and $t\ge 1$

$$\begin{aligned} \Vert P^*_t\mu -\nu \Vert _{\mathrm {TV}}\, \le \lambda ^{\lfloor {t}\rfloor }\Vert \mu -\nu \Vert _{\mathrm {TV}(\mathcal {C}^{\alpha _0}_\mathfrak {m})}. \end{aligned}$$

(5.30)

Proof

Using the existence of invariant measures, shown in Theorem 5.6, the proof of uniqueness follows exactly as that of [34, Cor. 6.6]. $\square$

We now complete the proof of Theorem 1.4.

Proof of Theorem 1.4

Let $(\alpha _0,\,\alpha ,\,\eta )$ satisfy the criteria of (1.3), the less restrictive set of parameters, and $(\alpha ',\,\alpha _0',\,\eta ')$ satisfy the more restrictive set, (5.6), and be such that $\alpha _0' >\alpha _0$. Then applying Corollary 5.19 with $(\alpha _0',\,\alpha ',\,\eta ')$ we find that there exists a unique invariant measure $\nu \in \mathcal {P}(\mathcal {C}^{\alpha }_\mathfrak {m})$ of $(P_t)_{t\ge 0}$ and that $\nu$ has full support in $\mathcal {C}^{\alpha }_\mathfrak {m}$. From Theorem 1.1, given $\zeta \in \mathcal {C}^{\alpha _0}(\mathbb {T})$, for any $t>0$, $u_t(\zeta )\in \mathcal {C}^{\alpha }_\mathfrak {m}(\mathbb {T})\subset \mathcal {C}^{\alpha _0'}_{\mathfrak {m}}(\mathbb {T})$. Therefore, setting $\mu = \mathcal {L}(u_t(\zeta ))$, in Corollary 5.19, we have, for some $\lambda \in (0,1)$,‘

$$\begin{aligned} \Vert P_{2t}^*\delta _\zeta - \nu \Vert _{\mathrm {TV}(\mathcal {C}^{\alpha })}&= \Vert P^*_{t} \mathcal {L}\left( u_{t}(\zeta )\right) -\nu \Vert _{\mathrm {TV}(\mathcal {C}^{\alpha _0})}\\&\le \lambda ^{\lfloor {t}\rfloor } \Vert \mathcal {L}\left( u_{t}(\zeta )\right) -\nu \Vert _{\mathrm {TV}(\mathcal {C}^{\alpha '_0})} \\&\le \lambda ^{\lfloor {t}\rfloor }, \end{aligned}$$

where in the penultimate line we used that $\mathcal {C}_b(\mathcal {C}^{\alpha _0}(\mathbb {T})) \subset \mathcal {C}_b(\mathcal {C}^{\alpha _0'}(\mathbb {T}))$ and in the last line that the total variation distance is bounded by 1. Therefore (1.4) holds for $c>0$ sufficiently small.

Consider now $p\ge 1$. By Fernique’s theorem ([6, Thm. 2.6]), there exists $\Lambda >0$ such that $\mathbb {E}[\exp (\Lambda \Vert v\Vert _{C_1\mathcal {C}^\alpha }^2)]<\infty$. Since $\mathcal {C}^\alpha \hookrightarrow L^p$ and $\Vert u_1(\zeta )\Vert _{L^p} \lesssim 1+\Vert v\Vert _{C_1\mathcal {C}^\alpha }^{1/\alpha }$ by Theorem 4.4, there exists $\Lambda >0$ such that

$$\begin{aligned} \sup _{\zeta \in \mathcal {C}^{\alpha _0}} \mathbb {E}\left[ \exp \left( \Lambda \Vert u_1(\zeta )\Vert ^{2\alpha }_{L^p}\right) \right] <\infty \;. \end{aligned}$$

(5.31)

Hence, taking $\alpha$ arbitrarily close to $\frac{1}{2}$ (after possibly restarting the equation and choosing a new $\alpha _0$ close to 0), we obtain (1.5). $\square$

Appendices

Appendix A: Hölder–Besov spaces on $\mathbb {T}$

For $\alpha \in \mathbb {R}$ and $p,\,q \in [1,\infty ]$ we define the non-homogeneous Besov norm for $f\in C^\infty (\mathbb {T})$ by the expression

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}^\alpha _{p,\,q}(\mathbb {T})} := \left\| \left( 2^{\alpha k}\Vert \Delta _k f\Vert _{L^p(\mathbb {T})}\right) _{k}\right\| _{l^q(\mathbb {Z})}, \end{aligned}$$

(A.1)

where $\Delta _k : L^1(\mathbb {T})\rightarrow L^1(\mathbb {T})$ are the Littlewood–Paley projection operators on the periodic, integrable functions. See for example [2, Ch. 2] for a construction on $\mathbb {R}^d$ and [21, App. A] for a discussion of the same construction in the periodic setting. We recall here that there exist kernels $h_k \in C^\infty (\mathbb {T})$ such that, for $f \in L^1(\mathbb {T})$ and $k\ge -1$, $\Delta _k f = h_k *f$ - see [2, Sec. 2.2]. Furthermore $\mathcal {F}h_k$, the Fourier transform of $h_k$, takes values in [0, 1] and is supported in the ball B(0, 2) for $k=-1$ and in the annulus $B(0,9\cdot 2^{k})\setminus B(0,2^{k})$ for $k\ge 0$.

We denote by $\mathcal {B}^\alpha _{p,\,q}(\mathbb {T})$ the completion of $C^\infty (\mathbb {T})$ with respect to (A.1), which ensures these spaces are separable.

Theorem A.1

(Duality Pairing—[21, Prop. A.1])

For $\alpha \in \mathbb {R}$, $p,\,q\in [1,\infty ]$ and $p',q' \in [1,\infty ]$ such that $\frac{1}{p}+ \frac{1}{p'} =\frac{1}{q}+\frac{1}{q'}=1$, there exists a constant $C:=C(\alpha ,p,q)>0$ such that

$$\begin{aligned} |\langle f,g\rangle | \,\le C \Vert f\Vert _{\mathcal {B}^\alpha _{p,q}}\Vert g\Vert _{\mathcal {B}^{-\alpha }_{p',q'}}. \end{aligned}$$

(A.2)

Theorem A.2

(Besov Embeddings - [21, Prop. A.2, Rem. A.3]) Let $\alpha \le \beta \in \mathbb {R}$, $q_1,\,q_2 \in [1,\infty ]$, $p\ge r \in [1,\infty ]$ be such that $\beta = \alpha +\left( \frac{1}{r}-\frac{1}{p} \right)$ and $q_1\ge q_2$. Then there exists a constant $C:=C(\alpha ,p,r,q_1,q_2)>0$ such that

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}^\alpha _{p,q_1}}\le C \Vert f\Vert _{\mathcal {B}^\beta _{r,q_1}}, \quad \text {and}\quad \Vert f\Vert _{\mathcal {B}^{\alpha }_{p,q_1}}\le C \Vert f\Vert _{\mathcal {B}^{\alpha }_{p,q_2}}. \end{aligned}$$

(A.3)

Furthermore, for any $\beta >\alpha$ and $p,q\in [1,\infty ]$ the embedding $\mathcal {B}^\beta _{p,q}\hookrightarrow \mathcal {B}^{\alpha }_{p,q}$ is compact. Finally, there exists $C:=C(p)>0$ such that

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

(A.4)

Theorem A.3

(Effect of Derivatives - [21, Prop. A.5]) Let $\alpha \in \mathbb {R}$, $p,\,q\in [1,\infty ]$ and $k \in \mathbb {N}_{>0}$. Then there exists a constant $C:=C(k)>0$ such that for any $\sigma \in \mathbb {N}$

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

(A.5)

Theorem A.4

(Bounds in Terms of the Derivative - [21, Prop. A.6]) Let $\alpha \in (0,1]$ and $p,\,q \in [1,\infty ]$, where if $\alpha =1$ we impose $q=\infty$. Then there exists a constant $C:=C(\alpha ,p,q)>0$ such that

$$\begin{aligned} \Vert f\Vert _{\mathcal {B}^{\alpha }_{p,q}} \le C\left( \Vert f\Vert _{L^p}^{1-\alpha }\Vert \partial _x f\Vert ^\alpha _{L^p} + \Vert f\Vert _{L^p}\right) \end{aligned}$$

(A.6)

Theorem A.5

(Product Bounds - [21, Prop. A.7]) Let $p,q \in [1,\infty ]$, $\alpha ,\,\beta \in \mathbb {R}$ be such that $\alpha +\beta >0$ with $\beta >0$. Then for $\alpha >0$ and $\nu \in [0,1]$, there exists a constant $C:=C(\alpha ,\beta ,p,q)>0$ such that

$$\begin{aligned} \Vert fg\Vert _{\mathcal {B}^{\alpha }_{p,q}} \le C \Vert f\Vert _{\mathcal {B}^{\alpha }_{\frac{p}{\nu },q}}\Vert g\Vert _{\mathcal {B}^{\beta }_{\frac{p}{1-\nu },q}}\le C \Vert f\Vert _{\mathcal {B}^{\alpha +(1-\nu )\frac{d}{p}}_{p,q}}\Vert g\Vert _{\mathcal {B}^{\beta +\nu \frac{d}{p}}_{p,q}}. \end{aligned}$$

(A.7)

Furthermore, if $\alpha >0$ and $p_1,p_2,p_3,p_4\in [1,\infty ]$ are such that $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p_3}+\frac{1}{p_4}=\frac{1}{p}$, then there exists a $C:= C(\alpha ,p,q,p_1,p_2,p_3,p_4)>0$ such that

$$\begin{aligned} \Vert fg\Vert _{\mathcal {B}^{\alpha }_{p,q}} \le C\left( \Vert f\Vert _{L^{p_1}}\Vert g\Vert _{\mathcal {B}^{\alpha }_{p_2,q}} + \Vert f\Vert _{\mathcal {B}^{\alpha }_{p_3,q}}\Vert g\Vert _{L^{p_4}} \right) . \end{aligned}$$

(A.8)

1.1 Appendix A.1: Parabolic and elliptic regularity estimates

We define the action of the heat semi-group on $f \in L^1(\mathbb {T})$ by,

$$\begin{aligned} e^{t\Delta }f := \mathcal {F}^{-1}\left( e^{-4\pi ^2|\,\cdot \,|^2t} \hat{f}_{\,\cdot \,}\right) = \mathcal {H}_t*f, \quad t>0, \end{aligned}$$

where $\mathcal {F}^{-1}$ denotes the inverse Fourier transform on $L^1(\mathbb {T})$ and

$$\begin{aligned} \mathcal {H}_t(x) := \frac{1}{\sqrt{4\pi t}}\sum _{n\in \mathbb {Z}} e^{-\frac{|x-n|^2}{4t}} \mathbbm {1}_{(0,\infty )}(t)= \sum _{m\in \mathbb {Z}} e^{-4\pi ^2|m|^2t}e_{m}(x)\mathbbm {1}_{(0,\infty )}(t), \end{aligned}$$

(A.9)

defines the heat kernel on $(0,\infty )\times \mathbb {T}$. We refer to [22, Prop. 5 & 6] for a proof of the following theorem.

Theorem A.6

(Regularising Effect of the Heat Flow - [22, Prop. 5 & 6]) Let $\alpha ,\,\beta \in \mathbb {R}$, $p\ge r \in [1,\infty ]$ and $q\in [1,\infty ]$. Then, if $\beta \le \alpha \le \beta +2$, there exists a constant $C:=C(\alpha ,\beta ,p,r,q)>0$, such that, uniformly over $t>0$,

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

(A.10)

Secondly, if $0\le \beta -\alpha \le 2$ then there exists a $C:=C(\alpha ,\beta ,p,q)>0$ such that for a any $t>0$,

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

(A.11)

Remark A.7

Since we have defined the Hölder–Besov spaces as the closure of $C^\infty (\mathbb {T})$ under the appropriate norm, for any $\alpha \in \mathbb {R}$, $p,\,q \in [1,\infty ]$, if $f\in \mathcal {B}^{\alpha }_{p,q}(\mathbb {T})$, we have that

$$\begin{aligned} \lim _{t\rightarrow 0}\Vert (1-e^{t\Delta })f\Vert _{\mathcal {B}^{\alpha }_{p,q}} = 0. \end{aligned}$$

The equivalent elliptic regularity estimate is an easy consequence of [2, Lem. 2.2] concerning Fourier multipliers.

Theorem A.8

Let $f\in C^\infty (\mathbb {T})$ be such that $\langle f,1\rangle =0$. Suppose $-\partial _{xx} \rho = f$ and $\langle \rho ,1\rangle =0$. Then for any $\alpha \in \mathbb {R}$, $p,q\in [1,\infty ]$, there exists a constant $C\ge 0$ such that

$$\begin{aligned} \Vert \rho \Vert _{\mathcal {B}^{\alpha }_{p,q}} \le C \Vert f\Vert _{\mathcal {B}^{\alpha -2}_{p,q}}. \end{aligned}$$

(A.12)

Proof

From [2, Lem. 2.2], there exists a $C>0$ such that for any $k\ge 0$ we have $\Vert \Delta _k \rho \Vert _{L^p} \le C2^{-2k}\Vert \Delta _kf\Vert _{L^p}$. Since $(\mathcal {F}f)(0)=(\mathcal {F}\rho )(0)=0$, $\Delta _{-1} \rho (\zeta ) = \Delta _{-1}f(\zeta )$. Therefore, there exists a dimension dependent constant $C>0$, such that for any $k\ge -1$,

$$\begin{aligned} 2^{\alpha k} \Vert \Delta _k\rho \Vert _{L^p} \le C 2^{(\alpha -2)k}\Vert \Delta _k f \Vert _{L^p}, \end{aligned}$$

from which the claim follows. $\square$

Remark A.9

Applying Theorem A.8 followed by Theorem A.3 gives us the estimate, for $f,\rho , \alpha ,p,q$ as in Theorem A.8 and $r\in [1,p]$,

$$\begin{aligned} \Vert \partial _x \rho \Vert _{\mathcal {B}^{\alpha }_{p,q}} \lesssim \Vert f\Vert _{\mathcal {B}^{\alpha -1+\left( \frac{1}{r}-\frac{1}{p}\right) }_{r,q}}. \end{aligned}$$

(A.13)

Appendix B: Regularity of the stochastic heat equation

We outline the main steps necessary to prove Theorem 2.3 using the characterisation of Hölder–Besov spaces. We note that a similar result can be proved using a version of the Garsia–Rodemich–Rumsey lemma, [14]. A more detailed presentation of similar results in our context can be found in [22, 34]. We include a proof directly in our setting for the reader’s convenience. A central tool is the following Kolmogorov regularity result which can be found in a similar form as [22, Lem. 9 & 10]. We recall the kernels $h_k \in C^\infty (\mathbb {T})$ from the beginning of “Appendix A”.

Lemma B.1

(Kolmogorov Regularity—[22, Lem. 9 & 10]) Let $(t,\phi ) \mapsto Z_t(\phi )$ be a map from $\mathbb {R}_+ \times L^2(\mathbb {T})\rightarrow L^2(\Omega ,\mathcal {F},\mathbb {P})$ that is linear and continuous in $\phi$ and such that $Z_0=0$. Assume that for some $p>1$, $\alpha '\in \mathbb {R}$, $\kappa '>\frac{1}{p}$ and $T>0$, there exists a constant $C>0$, such that for all $k\ge -1$, $x\in \mathbb {T}$ and $s,\,t \in [0,T]$,

$$\begin{aligned} \mathbb {E}\left[ |Z_t(h_k(\,\cdot \,-x)) - Z_s(h_{k}(\,\cdot \,-x))|^p \right] \le C^p |t-s|^{\kappa ' p} 2^{-k\alpha 'p}. \end{aligned}$$

(B.1)

Then, for all $\alpha <\tilde{\alpha }'-\frac{1}{p}$ and $\kappa \in [0,\kappa '-\frac{1}{p})$, there exists a modification $\tilde{Z}$ of Z such that

$$\begin{aligned} \mathbb {E}\left[ \Vert \tilde{Z}\Vert ^p_{\mathcal {C}^\kappa _T\mathcal {B}^{\alpha -\kappa }_{p,p}} \right] < \infty . \end{aligned}$$

Proof

It suffices to modify the proof of [22, Lem. 9 & 10] to our finite volume and one dimensional setting. $\square$

Proof of Theorem 2.3

Below we show that $\mathcal {L}(\mathcal {I}_{t_0,t})$ depends only on $|t-t_0|$ so without loss of generality we set $t_0=0$. It is also clear, by translation invariance of $\mathcal {H}$ in space and the law of the white noise, that $\mathcal {I}_{t_0,t}$ is translation invariant, i.e $\mathcal {I}_{t_0,t}(\phi (\,\cdot \,+x)) \sim \mathcal {I}_{t_0,t}(\phi (\,\cdot \,))$ for any $\phi \in C^\infty (\mathbb {T})$ and $x\in \mathbb {T}$.

For $k\ge -1$, $t \in (0,T]$ and setting $x=0$ in (B.1), we have $v_{0,t}(h_k)=\mathcal {I}_{0,t}(h_k)$. So splitting the increment using the semi-group property, Parseval’s theorem, the covariance formula for the space-time white noise and the action of the heat kernel in Fourier space we have for all $0\le s < t \le T$

$$\begin{aligned} \mathbb {E}\left[ |\mathcal {I}_{0,t}(h_k)-\mathcal {I}_{0,s}(h_k)|^2 \right]&\lesssim \sum _{\begin{array}{c} m \in \mathrm {supp}(\mathcal {F}h_k)\\ m\ne 0 \end{array}} \frac{1}{8\pi ^2|m|^2}(1-e^{-(t-s)8\pi ^2 |m|^2}) \\&\lesssim \sum _{\begin{array}{c} m \in \mathrm {supp}(\mathcal {F}h_k)\\ m\ne 0 \end{array}} \frac{|t-s|^\gamma }{(8\pi ^2|m|^2)^{1-\gamma }}\\&\lesssim |t-s|^\gamma 2^{-k(1-2\gamma )}, \end{aligned}$$

where $\mathcal {F}h_k$ denotes the Fourier transform of $h_k$ and we may choose any $\gamma \in [0,1]$. So then using Nelson’s estimate [24, Sec. 1.4.3], for any $p\ge 2$, there exists a constant $C:=C(p,T)>0$ such that for all $k\ge -1$ and $\gamma \in [0,1)$,

$$\begin{aligned} \mathbb {E}\left[ |\mathcal {I}_{0,t}(h_k)-\mathcal {I}_{0,s}(h_k)|^p \right] \le C |t-s|^{\frac{\gamma }{2}p} 2^{-k\left( \frac{1}{2}-\gamma \right) p}. \end{aligned}$$

Then applying Theorem B.1 and the Besov embedding (A.3), we see that there exists a modification of $t\mapsto v_{0,t}$ (which we do not relabel) such that for any $p\ge 1$, $\alpha \in (0,1/2),\,\kappa \in \left[ 0,1/2\right)$ and $T>0$,

$$\begin{aligned} \mathbb {E}\left[ \Vert v_{0,\,\cdot \,}\Vert ^p_{\mathcal {C}^{\kappa }_T \mathcal {C}^{\alpha -2\kappa } }\right] < \infty . \end{aligned}$$

To see stationarity of the processes $t_0\mapsto v_{t_0,t_0+h}$ for fixed $h>0$, we use the fact that $v_{t_0,t_0+h}(\phi )$ is Gaussian, and has zero mean, so that its law is entirely determined by its second moment. Therefore, letting $\phi \in C^\infty (\mathbb {T})$, and by similar computations as above, we see that

$$\begin{aligned} \mathbb {E}\left[ v_{t_0,t_0+h}(\phi )^2 \right]&= \sum _m \frac{1}{8\pi ^2|m|^2}\left( 1- e^{-8\pi ^2|m|^2h}\right) |\hat{\phi }_m|^2, \end{aligned}$$

and hence $\mathcal {L}(v_{t_0,t_0+h})$ depends only on $h\in [t_0,T-t_0]$. Finally, by the assumption that $\xi$ is spatially mean free, for any $t_0\in [0,T)$ and $h\in (T-t_0,T]$, $\mathcal {L}(v_{t_0,t_0+h}) \in C^\alpha _0(\mathbb {T})$. $\square$