Abstract
We consider the stochastic heat equation on \([0,\,1]\) with periodic boundary conditions and driven by space-time white noise. Under various natural conditions, we study small ball probabilities for the Hölder semi-norms of the solutions, and provide near optimal bounds on these probabilities. As an application, we prove a support theorem in these Hölder semi-norms.
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1 Introduction and main results
We consider the stochastic heat equation (SHE) on \({\mathbf {T}}:=[0,1]\) with periodic boundary condition and driven by space-time white noise (we identify \({\mathbf {T}}\) as the one-dimensional torus, i.e., \({\mathbf {T}}:={\mathbf {R}}/{\mathbf {Z}}\)). This is the real-valued random field \(u(t,x),\; t\in {\mathbf {R}}_+,\, x\in {\mathbf {T}}\) which solves
with given initial profile \(u(0,\cdot )=u_0: {\mathbf {T}}\rightarrow {\mathbf {R}}\) and satisfying \(u(t,0)=u(t,1)\) for all \(t\in {\mathbf {R}}_+\). The space-time white noise \(\dot{W}\) is a centered generalized Gaussian random field with \({\mathbb {E}}[\dot{W}(t,x) \dot{W}(s,y)]= \delta _0(x-y)\delta _0(t-s)\). We will make the following two assumptions on the function \(\sigma : {\mathbf {R}}_+\times {\mathbf {T}}\times {\mathbf {R}}\rightarrow {\mathbf {R}}\).
Assumption 1.1
The function \(\sigma \) is uniformly elliptic, that is, there are constants \({\mathscr {C}}_1>0\) and \({\mathscr {C}}_2>0\) such that
Assumption 1.2
The function \(\sigma \) is Lipschitz continuous in the third variable, that is there is a constant \({\mathscr {D}}>0\) such that
The existence and uniqueness of solutions to (1.1) under the above assumptions are well known. See for example [6] or [18] for the proofs and various other properties. It is also known that the solutions are Hölder(\(\frac{1}{2}-\)) in space and Hölder(\(\frac{1}{4}-\)) in time.
In this paper we study the probabilities of the events that the SHE (1.1) is unusually regular, as measured in certain Hölder semi-norms, up to a fixed time. To the best of our knowledge, our paper is the first to carry out such a study even though regularity properties of SPDEs have been very well studied. See for instance [7] and [17] where very precise information about the modulus of continuity is given.
Our study will be framed as small ball probabilities of these semi-norms. Small ball probabilities have been very well studied in many different settings but most of the results in the literature are for Gaussian processes; see [13] for an extensive survey on various developments and applications of small ball probabilities for Gaussian processes. However there has not been much progress for non-Gaussain processes. Only a handful of papers have looked at these types of questions for SPDEs. The paper closest to ours is that of the very recent [1] where the sup norm is considered. In another recent paper [11], heat equations with additive noise are considered under different norms and in [14], the stochastic wave equation is studied.
To phrase our results precisely we need some notations. Fix \(0<\theta <\frac{1}{2}\) and a terminal time \(T>0\). For a function \(f:[0, T] \times {\mathbf {T}}\rightarrow {\mathbf {R}}\) and for every \(t\in [0, T]\) and \(x\in {\mathbf {T}}\), let
be the spatial Hölder semi-norm and let
be the temporal Hölder semi-norm. The Hölder(\(\frac{1}{2}-\)) regularity in space and the Hölder(\(\frac{1}{4}-\)) regularity in time of u imply that \(\sup _{t\in [0,T]} {\mathcal {H}}_t^{(\theta )} (u)\) and \(\sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )} (u)\) are finite a.s.
The above quantities provide a quantitative way of measuring regularity of functions. This is a natural measure; indeed the smaller \({\mathcal {H}}_t^{(\theta )}(f)\) is the closer \(f(t,\cdot )\) is to a constant function, and similarly for \({\mathscr {H}}_x^{(\theta )}(f)\). We investigate the probability that they are exceptionally small for solutions to the stochastic heat equations.
There have been a few papers which investigate small ball probabilities for Hölder norms and various Sobolev norms, see for example [2, 4, 8, 9] and the references in [13]. We emphasize that the above results are for Gaussian processes but in general the SPDEs that we consider here will be non-Gaussian. Another novel feature of our work is that we are able to obtain bounds on the probabilities that the \({\mathcal {H}}_t^{(\theta )}(f)\) (resp. \({\mathscr {H}}_x^{(\theta )}(f)\) ) are uniformly small in time (resp. space), whereas [4, 8, 9] find bounds on the small ball probabilities of the Hölder norms of Gaussian processes X(t) indexed by only one parameter \(t\in {\mathbf {R}}_+\).
Before we state the main result, we introduce one more notation. For each \(\theta \in (0,\, 1/2)\), we let \(\Lambda =\Lambda (\theta )\) be given by
where p(t, x) is the Gaussian density (2.1) and \(\Gamma (t)\) is the Gamma function.
Theorem 1.1
Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\). Suppose that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2}\left( 1\wedge \frac{1}{\Lambda }\right) \). Then for any \(\eta >0\) there exist positive constants \(C_1,C_2,C_3,C_4>0\) dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2,{\mathscr {D}}, \theta ,\eta \) such that
One can improve the lower and upper bounds in (1.5) by imposing more restrictions on \(\sigma \); see Theorems 1.2, 1.3 and 1.4 below, and Remark 6.1. In Sect. 8 below we provide several support theorems where bounds on the probability that u is close (in the above Hölder semi-norm) to a function h in certain classes (such as Hölder spaces) are provided.
The reader might wonder whether the upper and lower bounds in (1.5) hold when we consider the Hölder norm
instead of semi-norm considered in (1.5), where \(\Vert u\Vert _{\infty , T}:=\sup _{(t,x) \in [0,T]\times {\mathbf {T}}} |u(t,x)|\). This is not the case. Indeed, due to the 1 : 2 : 4 scaling of fluctuations, space and time for the SHE, the Hölder semi-norm of the SHE in time-space regions of the form \([t, t+\epsilon ^{\frac{2}{\theta }}] \times [x, x+\epsilon ^{\frac{1}{\theta }} ]\) fluctuates by order \(\epsilon \). However in these regions, the solution u itself fluctuates by order \(\epsilon ^{\frac{1}{2\theta }}\). Intuitively, what we try to show in this article is that the \(T\epsilon ^{-\frac{3}{\theta }}\) time-space boxes obtained by dividing \([0,T]\times {\mathbf {T}}\) into subintervals of the form \([t, t+\epsilon ^{\frac{2}{\theta }}] \times [x, x+\epsilon ^{\frac{1}{\theta }} ]\) can be somewhat viewed as independent regions. This explains the \(T\epsilon ^{-\frac{3}{\theta }}\) that we obtain in the exponents in (1.5). Moreover, by this reasoning, one should expect similar bounds on the probability \(P(\Vert u\Vert _{\infty , T} \le \epsilon ^{\frac{1}{2\theta }} )\) if we start at \(u_0\equiv 0\), for example. In fact this is what was proved in [1]. Therefore, while it is not true that we have the same bounds as (1.5) for the Hölder norm (1.6), we do have the same bounds for
if we start with \(u_0\) such that \(\Vert u_0\Vert _{\infty } \le \frac{\epsilon ^{\frac{1}{2\theta }} }{2}\) and \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2}\left( 1\wedge \frac{1}{\Lambda }\right) \).
1.1 Results
Instead of looking at the probability of the event in (1.5) directly, we consider the probabilities of the events \(\left\{ \sup _{t\in [0,T]} {\mathcal {H}}_t^{(\theta )} (u)\le \epsilon \right\} \) and \(\left\{ \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )} (u)\le \epsilon \right\} \) separately. The bounds in (1.5) will then follow from the bounds on the probabilities of these two events (see Sect. 6). It turns out that the regularity in time (as measured by the smallness of \({\mathscr {H}}_x^{(\theta )}(u)\)) is intimately connected to the regularity in space (as measured by the smallness of \({\mathcal {H}}_t^{(\theta )}(u)\)). Our arguments indicate that for the solution to be regular in time it is necessary for it to be regular in space. We now state the small ball probability estimates for \(\sup _{t\le T} {\mathcal {H}}_t^{(\theta )}(u)\) and \(\sup _{x\in {\mathbf {T}}} {\mathscr {H}}^{(\theta )}(u)\) with varying assumptions on the nonlinearity \(\sigma \).
Theorem 1.2
Assume that the function \(\sigma (t,x,u)\) is independent of u and satisfies Assumption 1.1. Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\).
-
(a)
Suppose that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2}\). Then there exist positive constants \(C_1, C_2,C_3,C_4 >0\) dependent only on \({\mathscr {C}}_1,{\mathscr {C}}_2,\theta \) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }}}\right) \le P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}}\right) . \end{aligned}$$(1.7) -
(b)
Suppose that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2\Lambda }\). Then there exist positive constants \(C_1, C_2,C_3,C_4 >0\) dependent only on \({\mathscr {C}}_1,{\mathscr {C}}_2,\theta \) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }}}\right) \le P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}}\right) . \end{aligned}$$(1.8)
It can be shown that u is a Gaussian random field when \(\sigma \) does not depend on u. The proof of the above theorem takes up a significant part of this paper and hinges on well known results specific to Gaussian processes, as well as the recently proved Gaussian correlation inequality ([16]). We next consider the case when \(\sigma \) can also depend on u.
Theorem 1.3
Suppose that \(\sigma (t,x,u)\) satisfies both Assumptions 1.1 and 1.2. Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\).
-
(a)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2}\). Then for any \(\eta >0\) there exist positive constants \(C_1,C_2,C_3,C_4>0\) dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2,{\mathscr {D}}, \theta ,\eta \) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }+\eta }}\right) \le P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$(1.9) -
(B)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2\Lambda }\). Then for any \(\eta >0\) there exist positive constants \(C_1,C_2,C_3,C_4>0\) dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2, {\mathscr {D}}, \theta ,\eta \) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }+\eta }}\right) \le P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$(1.10)
Our next result sharpens the lower bound of the above theorem by imposing a further restriction on the Lipschitz coefficient of \(\sigma \).
Theorem 1.4
Suppose that \(\sigma (t,x,u)\) satisfies both Assumptions 1.1 and 1.2. Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\).
-
(a)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2}\). Then there is a \({\mathscr {D}}_0>0\) such that for all \({\mathscr {D}}<{\mathscr {D}}_0\), there exist positive constants \(C_1, C_2, C_3, C_4 >0\) dependent only on \({\mathscr {C}}_1,{\mathscr {C}}_2,\theta \) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2T}{\epsilon ^{\frac{3}{\theta }}}\right) \!\le \! P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u\right) \le \epsilon \right) \!\le \! C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) .\qquad \end{aligned}$$(1.11) -
(b)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2\Lambda }\). Then there is a \({\mathscr {D}}_1>0\) such that for all \({\mathscr {D}}<{\mathscr {D}}_1\), there exist positive constants \(C_1, C_2, C_3, C_4 >0\) dependent only on \({\mathscr {C}}_1,{\mathscr {C}}_2,\theta \) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }}}\right) \le P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$(1.12)
We now say a few words about the proofs of our theorems. As mentioned above, Theorem 1.2 relies heavily on the fact that when \(\sigma \) is independent of u, the solution u(t, x) is a Gaussian random field. The proof of the upper bound is essentially contained in the Proof of Lemma 3.2 which among other things, relies on the Gaussianity of u. Another crucial element is the sharp bound given by Lemma 2.5 whose proof uses some well known ideas presented in [1]. It is also interesting to note that when \(\sigma \) is a constant, one can further simplify the proof of the upper bound by resorting to Slepian’s lemma; see Remark 3.4 for more details. The lower bounds rely even more heavily on Gaussianity of the solution in that we use the Gaussian correlation inequality in an essential way. This is done in Lemma 4.2. Another key ingredient is the use of a change of measure argument similar to [1]. Intuitively, this allows us to keep the solution small which gives us a better handle on the estimates required.
Under the conditions of Theorems 1.3 and 1.4, the solutions are no longer Gaussian, so the corresponding proofs require different strategies. For the lower bounds, we use a perturbation argument together with the proof of the lower bound in Theorem 1.2. We note that the sharper lower bound in Theorem 1.4 is also a consequence of the very same perturbation argument.
The proofs of the upper bounds in Theorems 1.3 and 1.4 are entirely different and make use of certain auxiliary random variables which have nice independence properties. These random variables are indexed by the spatial variables. Their construction is inspired by [5].
In the final section of this paper, we present some extensions and prove a support theorem in the Hölder semi-norm. It will be clear later that our paper raises several questions. One such open question is whether the bounds (1.7) and (1.8) continue to hold in the general case, that is for any \(\sigma \) satisfying Assumptions 1.1 and 1.2. We have assumed that \(\sigma \) is bounded below and above by positive constants. Another avenue of investigation is to replace these assumptions by less stringent ones. Let us point that here, when \(\theta =\frac{1}{2}\) the above theorems match the results recently obtained in [1] for the small ball probabilities of the sup norm of u.
We have studied the small ball probability estimates of \(\sup _{t\in [0,T]} {\mathcal {H}}_t^{(\theta )} (u)\) and \(\sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )} (u)\). We next consider the small ball probability estimates of \({\mathcal {H}}_T^{(\theta )}(u)\) for a fixed time T, and \({\mathscr {H}}_X^{(\theta )}(u)\) for a fixed spatial point X. We start with the Gaussian case.
Theorem 1.5
Assume that the function \(\sigma (t,x,u)\) is independent of u and satisfies Assumption 1.1, and fix a time \(T>0\). Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\).
-
(a)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2}\). Then there exist positive constants \(C_1, C_2,C_3,C_4 >0\) dependent only on \({\mathscr {C}}_1,{\mathscr {C}}_2,\theta , T\) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 }{\epsilon ^{\frac{1}{\theta }}}\right) \le P \left( {\mathcal {H}}_T^{(\theta )}\left( u\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4}{\epsilon ^{\frac{1}{\theta }}}\right) . \end{aligned}$$(1.13) -
(b)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2\Lambda }\). Then there exist positive constants \(C_1, C_2,C_3,C_4 >0\) dependent only on \({\mathscr {C}}_1,{\mathscr {C}}_2,\theta \) such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T }{\epsilon ^{\frac{2}{\theta }}}\right) \le P \left( {\mathscr {H}}_X^{(\theta )}\left( u\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4 T}{\epsilon ^{\frac{2}{\theta }}}\right) . \end{aligned}$$(1.14)
The upper bounds are in fact an immediate consequence of the Proof of Theorem 1.2, and we will see that the constants \(C_3\) and \(C_4\) can be chosen independently of T. The lower bounds follow from the arguments in the proof of Theorem 2.2. of [9]. The proof of the lower bound above is specific to Gaussian processes and cannot be directly extended to the general case.
Theorem 1.6
Suppose that \(\sigma (t,x,u)\) satisfies both Assumptions 1.1 and 1.2, and fix a time \(T>0\). Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\).
-
(a)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2}\). Then there exist positive constants \(C_1,C_2>0\) dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2,\theta , T\) such that
$$\begin{aligned} P \left( {\mathcal {H}}_T^{(\theta )}\left( u\right) \le \epsilon \right) \le C_1\exp \left( - \frac{C_2}{\epsilon ^{\frac{1}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$(1.15) -
(b)
Assume that the initial profile satisfies \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{2\Lambda }\). Then there exist positive constants \(C_1,C_2>0\) dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2,\theta \) such that
$$\begin{aligned} P \left( {\mathscr {H}}_X^{(\theta )}\left( u\right) \le \epsilon \right) \le C_1\exp \left( - \frac{C_2T}{\epsilon ^{\frac{2}{\theta }}}\right) . \end{aligned}$$(1.16)
A trivial lower bound is obtained from either Theorem 1.3 or Theorem 1.4 depending on whether \(\sigma \) satisfies the assumptions of Theorem 1.3 or Theorem 1.4. However this is very far from the lower bound obtained in Theorem 1.5.
Remark 1.1
As the reader observes the bounds in Theorems 1.1, 1.3, 1.4 and 1.6 are close to optimal but not sharp. While optimal results can be obtained in the Gaussian case (i.e. when \(\sigma \) does not depend on u) using Gaussian-specific techniques, a perturbation argument is the main tool for the lower bounds in the non-Gaussian case. When the perturbation (as measured by the Lipschitz constant \({\mathscr {D}}\)) is small, one can get similar lower bounds as in the Gaussian case (see Theorem 1.4). However, the perturbation argument works only when the time interval under consideration is small and therefore does not work for Theorem 1.6. Moreover, a similar perturbation argument cannot be implemented for the upper bounds as we don’t have good control of the tail probabilities \(\left\{ |u(t,x)-u(s,x)|>\epsilon \right\} \) when \(|t-s|=O(\epsilon ^{\frac{2}{\theta }})\) (see Sect. 4.3 for the perturbation argument for lower bounds). We thus have to resort to a general non-Gaussian argument which only gives us close to optimal upper bounds.
Remark 1.2
The dependence of the constants in the theorems on the parameters \({\mathscr {C}}_1,{\mathscr {C}}_2,{\mathscr {D}} , \theta \) etc. is in general quite complicated. See for example Remark 3.3. We have tried to indicate the dependence on the parameters wherever we could.
Remark 1.3
Note that \(v(t,x) = u(\rho t, x)\) satisfies
for some other white noise \(\dot{{\tilde{W}}} \), and \({\tilde{\sigma }}(t,x,v) := \rho ^{\frac{1}{2}} \sigma (\rho t,x, v)\). The function \({\tilde{\sigma }}\) satisfies \(\rho ^{\frac{1}{2}}{\mathscr {C}}_1\le {\tilde{\sigma }}(t,x,v) \le \rho ^{\frac{1}{2}}{\mathscr {C}}_2\) and \(|{\tilde{\sigma }}(t,x,u) -{\tilde{\sigma }}(t,x,v)|\le \rho ^{\frac{1}{2}}{\mathscr {D}} |u-v|\). Thus one can obtain similar results with the inclusion of a diffusion parameter \(\rho \) by converting it to the form (1.1).
Remark 1.4
Note that it is sufficient to prove the above theorems for all sufficiently small \(\epsilon <\epsilon _0\), where \(\epsilon _0\) is dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta \) (and maybe additionally on \(\eta \) in the case of Theorem 1.3 and T in the case of Theorems 1.5 (a) and 1.6 (a)). The conclusion for any \(0<\epsilon <1\) follows from the fact that the probabilities \(P \left( {\mathcal {H}}_T^{(\theta )}\left( u\right) \le \epsilon \right) \), \(P \left( {\mathscr {H}}_X^{(\theta )}\left( u\right) \le \epsilon \right) \), \(P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u\right) \le \epsilon \right) \) and \(P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u\right) \le \epsilon \right) \) are nondecreasing in \(\epsilon \).
Remark 1.5
In this paper, we have studied bounds on various small ball probabilities. A related question would be to study the small ball constant, that is to find the constant C in the following
under appropriate conditions on \(\sigma \). These types of questions are challenging and are beyond the scope of this current paper. Using the techniques in [12], we are confident that one can successfully tackle the small ball constant problem in some cases, but we leave this for future work.
Plan: Sect. 2 contains some preliminary estimates. The proofs of the upper bounds in Theorems 1.2 and 1.3 are given in Sect. 3, while the correponding lower bounds are given in Sect. 4. The Proof of Theorem 1.4 is given in Sect. 5. After this, we give the Proof of Theorem 1.1 in Sect. 6. The proofs of Theorems 1.5 and 1.6 are presented in Sect. 7. Finally in Sect. 8, we give some extensions and prove a support theorem as a corollary of our results.
The following table highlights the main differences between the main theorems and their extensions (Table 1).
Notation: Throughout this paper, C with or without subscripts will denote positive constants whose value might change from line to line. Unless mentioned otherwise they will be independent of the parameters \(\epsilon , {\mathscr {C}}_1, {\mathscr {C}}_2, {\mathscr {D}}\) etc. We will sometimes emphasize that the dependence of the constants on specific parameters will be denoted by specifying the parameters in brackets, e.g. \(C(\delta )\). For a random variable X we denote \(\Vert X\Vert _p:=E[|X|^p]^{1/p}\).
2 Preliminaries
We define the heat kernel G(t, x) as the fundamental solution of the heat equation on the torus \({\mathbf {T}}\)
Let
be the fundamental solution of the heat equation on \({\mathbf {R}}\). It is known that the heat kernel on \({\mathbf {T}}\) is given explicitly by
We interpret the solution to (1.1) in the sense of Walsh ([18]) as a random field which satisfies
for each t and x, where the first term on the right is the space convolution of the heat kernel with the initial profile \(u_0(x)\), i.e.,
and the second term which we call the noise term is the space-time convolution of the heat kernel with the product of \(\sigma \left( s, y, u(s,y)\right) \) and white noise:
We are working on the torus \({\mathbf {T}}:={\mathbf {R}}/{\mathbf {Z}}\), so in the above two expressions \(x-y\) should be interpreted as the unique point z in [0, 1) such that \(x-y=z+k\) for some \(k\in {\mathbf {Z}}\).
We now show that it is enough to prove our main results under the assumption that \(u_0\equiv 0\). For a function \(g: {\mathbf {T}}\rightarrow {\mathbf {R}}\) dependent only on the spatial variable x, define
(Note the absence of subscript t in \({\mathcal {H}}^{(\theta )}\)). The first lemma is a simple observation about the spatial Hölder regularity of \(G_t*u_0\).
Lemma 2.1
If for some \(a>0\) one has \({\mathcal {H}}^{(\theta )}(u_0)\le a\) then \({\mathcal {H}}^{(\theta )}\big (G_t*u_0\big ) \le a\) for all \(t>0\).
Proof
Let \({\tilde{u}}_0:{\mathbf {R}}\rightarrow {\mathbf {R}}\) be the periodization of \(u_0\), that is \({\tilde{u}}_0(x+k)=u_0(x)\) for all \(k \in {\mathbf {Z}}\) and \(x\in {\mathbf {T}}\). We have
We have \(\left| {\tilde{u}}_0(x-w)-{\tilde{u}}_0(y-w) \right| \le a |x-y|^{\frac{1}{2}-\theta } \) by assumption and \(p(t,\cdot )\) integrates to 1, therefore the result follows. \(\square \)
We now prove a similar result for the temporal Hölder regularity of \((G_{\cdot } *u_0)(x)\). Recall the constant \(\Lambda \) introduced in (1.4).
Lemma 2.2
If \({\mathcal {H}}^{(\theta )}(u_0)\le a\) then
Proof
Without loss of generality assume that \(s<t\). Let \(g(x) = \left( G_s*u_0\right) (x)\) and \({\tilde{g}}\) be the periodization of g. Then by arguments similar to Lemma 2.1 we have
by a simple change of variables. \(\square \)
Now consider the random field
where u(t, x) solves (1.1) with the initial profile \(u_0\). One can easily check that
with initial profile \(v_0\equiv 0\), where
Furthermore \(|{\widetilde{\sigma }}(t,x,v) -{\widetilde{\sigma }}(t,x,w)|\le {\mathscr {D}}|v-w|\) and \({\widetilde{\sigma }}\) is bounded below and above by \({\mathscr {C}}_1, {\mathscr {C}}_2\).
Assume \({\mathcal {H}}^{(\theta )}(u_0) \le \frac{\epsilon }{2}\). Then from Lemma 2.1 and (2.6), we have the following implications:
Similar implications hold when we just consider \({\mathcal {H}}_T^{(\theta )}(u)\) and \({\mathcal {H}}_T^{(\theta )}(v)\) (without the supremum in t).
Similary if \({\mathcal {H}}^{(\theta )}(u_0) \le \frac{\epsilon }{2\Lambda }\), then from Lemma 2.2,
Similar implications hold when we just consider \({\mathscr {H}}_X^{(\theta )}(u)\) and \({\mathscr {H}}_X^{(\theta )}(v)\) (without the supremum in x).
Remark 2.1
(Important) From the above discussion we observe that it is sufficient to prove the main theorems stated in the introduction with \(u_0\equiv 0\). This is of course not surprising since the Laplacian is known to have smoothing effects. The above argument is merely a weak manifestation of this. We will assume that the initial profile \(u_0\equiv 0\) for the rest of the article.
We will need the following lemmas which were proved in [1].
Lemma 2.3
(Lemma 3.3 in [1]) There exist constants \(C_1, C_2\) such that for all time points \(0<s<t<1\), spatial points \(x, y\in {\mathbf {T}}\), and \(\lambda >0\),
Lemma 2.4
(Lemma 3.4 in [1]) There exist universal constants \({\mathbf {K}}_1, \, {\mathbf {K}}_2>0\) such that for all \(\alpha , \lambda , \epsilon >0\), \(\theta >0\), and for all \(a \in [0, 1)\) with \(a+\epsilon ^{1/\theta }<1\) we have
Remark 2.2
Note that Lemma 3.4 in [1] provides (2.7) when \(a=0\), however, one can follow exactly the same proof to get (2.7) for any \(a\in (0,1)\) with \(a+\epsilon ^{1/\theta }<1\). It was also pointed out in [1, Remark 3.1] that if \(\big |\sigma \big (s, y, u(s,y)\big )\big |\le C_1\epsilon ^{\frac{1}{2\theta }}\) then one can bound the right hand side of (2.7) by \(\frac{{\mathbf {K}}_1}{1\wedge \sqrt{\alpha }}\exp \left( - {\mathbf {K}}_2 \frac{\lambda ^2}{C_1^2\epsilon ^{\frac{1}{\theta }}\sqrt{\alpha }}\right) \).
The analysis of the following function will play a crucial role in this paper.
Although we have not made it explicit, the function \({\widetilde{N}}(t,x,y)\) clearly depends also on \(\theta \). The following lemma is used several times in the paper.
Lemma 2.5
Let \(\theta \in (0, \frac{1}{2})\). There exist constants \({\mathbf {K}}_3, {\mathbf {K}}_4\) dependent only on \(\theta \) such that for all \(\alpha , \lambda , \epsilon >0\) and for all \(a\in [0,1)\) with \(a+\epsilon ^{1/\theta }<1\), we have
Proof
We will show (2.9) when \(a=0\) and the same proof works for the general \(a\in (0,1)\), which is similar to Lemma 2.4 and Remark 2.2. Let us first consider the case when \(\alpha \ge 1\). Consider the following grid on \([0,\alpha \epsilon ^{\frac{2}{\theta }}]\times [0, \epsilon ^{\frac{1}{\theta }}]\), where the first coordinate is time and the second is space:
The grid \({\mathbb {G}}_n\) will consist of only the point (0, 0) if \(n<n_0\) where
Any \(p\in {\mathbb {G}}_n\) will be of the form \(\left( \frac{j}{2^{2n}}, \, \frac{k}{2^n}\right) \) , so for notational convenience, we set
We will choose two parameters \(0<\gamma _0(\theta )<\gamma _1(\theta )<\frac{1}{2}\) which depend only on \(\theta \) which satisfy the following constraint
We will fix the constant
and consider the events
By p, q being spatial neighbors in the grid \({\mathbb {G}}_n\), we mean that p, q have the same time coordinate but their spatial coordinates are adjacent in \({\mathbb {G}}_n\). For instance \(\left( \frac{j}{2^{2n}}, \, \frac{k-1}{2^n}\right) \) and \(\left( \frac{j}{2^{2n}}, \, \frac{k+1}{2^n}\right) \) are spatial neighbors of \(\left( \frac{j}{2^{2n}}, \, \frac{k}{2^n}\right) \) .
The number of such pairs of points p, q is bounded by \(2\cdot \alpha \epsilon ^{\frac{2}{\theta }}2^{2n} \cdot \epsilon ^{\frac{1}{\theta }} 2^n \le 2^4\cdot 2^{3(n-n_0)}\), where we have used (2.10). Therefore a union bound along with the first tail bound in Lemma 2.3 gives
where the second inequality follows since \(\alpha ^\theta \epsilon ^2 2^{2n_0\theta } \ge 1\) by our choice of \(n_0\) and the final inequality is obtained using the choice \(\gamma _0, \gamma _1\) in (2.11).
We now let \(A(\lambda ):= \cap _{n\ge n_0} A(n, \lambda )\) and use a union bound once again to obtain
Now on the event \(A(\lambda )\), one has for p, q spatial neighbors in \({\mathbb {G}}_n\)
by our choice of \(\gamma _0, \gamma _1\) and K in (2.11) and (2.12).
We now show that the above bound continues to hold when \(p, q\in {\mathbb {G}}_n\) are no longer spatial neighbors but have the same time coordinate. Let the spatial coordinate of p be \(k2^{-n}\) and let the spatial coordinate of q be \(l2^{-n}\), and without loss of generality assume \(k<l\). Find the smallest positive integer \(n_1\) with \(n_0\le n_1\le n\) such that
for some nonnegative integer \(k_1\). First note that we must have
The lower bound is clear by (2.13) and the upper bound follows from the minimality of \(n_1\), for if the difference between \(k2^{-n}\) and \(l2^{-n}\) was larger than \(2^{2-n_1}\) then there would be two spatial neighbors in \({\mathbb {G}}_{n_1-1}\) between them.
One next observes that we can find a sequence of points \(p_i, \, n_1\le i \le n\) and \(q_i,\, n_1\le i\le n\) with the same time coordinates as p, q, such that \(p_i, p_{i+1}\) (resp. \(q_i, q_{i+1}\)) are either equal or adjacent spatial points in \({\mathbb {G}}_i\). In addition at most one such adjacent spatial pair \((p_i, p_{i+1})\) (resp. \((q_i, q_{i+1})\)) is in each \({\mathbb {G}}_j,\, n_1\le j\le n\), and \(p_n=p, q_n=q\). Therefore
on the event \(A(\lambda )\). As a consequence, on this event
by (2.14) and our choice of \(\gamma _0,\gamma _1\) and K in (2.11) and (2.12). This completes the proof in the case \(\alpha \ge 1\).
In the case \(0<\alpha <1\), we divide the spatial interval into smaller intervals of length \(\sqrt{\alpha } \epsilon ^{\frac{1}{\theta }}\) to get that
Then, as explained in the beginning of the proof of this lemma (see also Remark 2.2), the probabilities inside the sum on the right hand side above have the same upper bound as for
We now apply the previous argument to finish the proof. \(\square \)
Remark 2.3
From Remark 2.2, it also follows from the proof that if \(\big |\sigma \big (s, y, u(s,y)\big )\big |\le C_1\epsilon ^{\frac{1}{2\theta }}\) then one can bound the right hand side of (2.9) by \(\frac{{\mathbf {K}}_3}{1\wedge \sqrt{\alpha }}\exp \left( - {\mathbf {K}}_4 \frac{\lambda ^2}{C_1^2\epsilon ^{\frac{1}{\theta }}\alpha ^\theta }\right) \).
Define
The proof of the following lemma is similar to that of Lemma 2.5 and is therefore omitted.
Lemma 2.6
Let \(\theta \in (0, \frac{1}{2})\). There exist constants \({\mathbf {K}}_7, {\mathbf {K}}_8\) dependent only on \(\theta \) such that for all \(\alpha , \lambda , \epsilon >0\) and for all \(a\in [0,1)\) with \(a+\epsilon ^{1/\theta }<1\), we have
Remark 2.4
Note also here that a similar statement to that of Remark 2.3 also holds in this case. That is, if \(\big |\sigma \big (s, y, u(s,y)\big )\big |\le C_1\epsilon ^{\frac{1}{2\theta }}\), then one can bound the right hand side of (2.7) by \(\frac{{\mathbf {K}}_7}{1\wedge \sqrt{\alpha }}\exp \left( - {\mathbf {K}}_8 \frac{\lambda ^2}{C_1^2\epsilon ^{\frac{1}{\theta }}\alpha ^\theta }\right) \).
We will also need some estimates concerning \(G(t,\,x)\), which come from Lemmas 3.1 and 3.2 of [1]. For the lemma below let
We have
Lemma 2.7
There exist positive constants \(C_0, C_1, {\tilde{C}}_1, C_2, C_3\) such that
With the preliminaries in place we can now move on to proving the theorems stated in the introduction.
3 Upper bounds
3.1 Upper bound in Theorem 1.2 (a)
We are assuming that the function \(\sigma (t,x,u)=\sigma (t,x)\) does not depend on the third variable so the random field u(t, x) is Gaussian. Before proving the required estimates, we describe the main strategy behind the proof.
Fix parameters \(c_0>0, c_1\ge 4\) to be specified later, and let
We consider discrete time-space points \((t_i, x_j)\), where the time points \(t_i\) are uniformly spaced in [0, T] and space points \(x_j\) are uniformly spaced in \({\mathbf {T}}\):
We clearly have
Consider the events
From the above
We will show in Lemma 3.2 below that for some \(0<\eta <1,\)
uniformly in i. Since the above bound holds regardless of the profile up to time \(t_{i-1}\) one can conclude that the right hand side of (3.4) is bounded by \(\eta ^{J(I+1)}\), which gives us the required upper bound in Theorem 1.2.
We have the following lemma which plays an important role along with the fact the solution is Gaussian. For \(k\in {\mathbf {N}}^+\) and \(\delta >0\), we define
Lemma 3.1
Fix \(c_0>0\) and \(c_1\ge 4\). Then there exist positive constants \(C_0,\,C_1,\, C_2\) such that for all \(\delta \) small enough,
uniformly in k. If \(0<|x_k-x_l|< \frac{1}{2}\) then
Proof
Since \({\tilde{\Delta }}_k\) is a mean zero random variable, we can use Itô’s isometry along with the bound on \(\sigma \) given by (1.2) to obtain
where the last inequality comes from Lemma 2.7. This gives the required upper bound in (3.7)
Next, we have
which gives the lower bound in (3.7). Let us explain how we obtain the last inequality above. For \(k\ge 1\) and \(\delta \) small enough one has \((k-\delta )^2\ge \frac{1}{10}(k^2+\delta ^2)\). Thus for all \(0\le s\le 2t_1\),
The second last inequality is a consequence of bounding the sum from above by an appropriate integral from a Riemann sum approximation.
We next turn to the bound on the covariance. Observe that if we assume that \(k>l\), we have \(a=x_k-x_l = (k-l)c_1\delta \) and therefore \(a+\delta>a-\delta >a/4\). Using this, the semigroup property of the heat kernel and (2.17) we have
\(\square \)
For the next lemma recall the events \(A_i\) defined in (3.3)
Lemma 3.2
Let \(c_0=1\) and \(c_1=K \frac{{\mathscr {C}}_2^3}{{\mathscr {C}}_1^3}\). We can find a \(K>0\) large enough and \(0<\eta <1\) such that for an arbitrary initial profile \(u_0\),
Remark 3.1
The above lemma implies (3.5) because the initial profile is allowed to be arbitrary, and by the Markov property, \(A_i\) depends only on the profile \(u(t_{i-1},\cdot )\).
Remark 3.2
(Important) Note that the arbitrariness of \(u_0\) assumed in the lemma is so that we can apply the Markov property in (3.5), since a priori there is no restriction on \(u(t_{i-1},\cdot )\). The Markov property is used several times in this paper and we shall use the arbitrariness of the profile at time \(t_{i-1}\) often. However the reader should not be confused with Remark 2.1 where we assumed that the initial profile (i.e. at time 0) for (1.1) is \(u_0\equiv 0\).
Proof
For an arbitrary initial profile \(u_0\)
where
We will show that each of the terms inside the product sign in (3.9) is uniformly (in j) bounded away from 1, which will imply the lemma. We will in fact prove a stronger statement that \(P\left( B_j \,\vert \, {\mathcal {G}}_{j-1}\right) \) is uniformly (in j) bounded away from 1, where \({\mathcal {G}}_{j-1}\) is the \(\sigma \) algebra generated by the random variables \({\tilde{\Delta }}_k = N(t_1, x_k+\delta ) - N(t_1, x_k), \, k\le j-1\). We thus need to show the existence of some \(0< \eta <1\) such that
where \( \Delta _k := u(t_1, x_k+\delta ) - u(t_1, x_k)\). We will obtain this by showing
for some constant C independent of j. We can use general properties of Gaussian random vectors to write
where
is the conditional expectation of \({\tilde{\Delta }}_j\) given \({\mathcal {G}}_{j-1}\). The variance of Y is the conditional variance of \({\tilde{\Delta }}_j\) given \({\mathcal {G}}_{j-1}\), which is also the conditional variance in (3.11). Moreover Y is independent of \({\mathcal {G}}_{j-1}\) and thus
Therefore for all \(l=0,1,\ldots , j-1\) we have
Let \({\mathbf {y}}= (y_0, y_1,\ldots , y_{j-1})^T\) where \(y_l\) represents the entry on the left hand side above, \(\varvec{\beta }\) be the vector of the \(\beta _l\)’s, and let
be the covariance matrix of the increments \({\tilde{\Delta }}_l\). We can thus rewrite (3.13) in matrix form
Let us next show that \(\mathbf{S}\) is invertible. Write \(\mathbf{S}=D(I-A)D\), where \({\mathbf {D}}\in {\mathbf {R}}^{j \times j}\) is the diagonal matrix with diagonal entries
and \(\mathbf{I}-A\) is the correlation matrix of \({\tilde{\Delta }}_l\). Above \(\text {Std}\) denotes the standard deviation of the random variable in parentheses. Denote by \(\Vert \cdot \Vert _{1,1}\) the norm on matrices in \({\mathbf {R}}^{j\times j}\) induced by the \(\ell _1\) norm \(\Vert \cdot \Vert _1\) on \({\mathbf {R}}^j\). Now \(\Vert \mathbf{A}\Vert _{1,1} = \max _j \sum _{i=1}^n |a_{i,j}|\) (see page 259 in [15]), we can use (3.7) and (3.8) to obtain
by choosing \(c_1=K \frac{{\mathscr {C}}_2^3}{{\mathscr {C}}_1^3}\) for a large K. In that case the inverse of \({\mathbf {I}}-{\mathbf {A}}\) exists and moreover
Using this along with the lower bound in (3.7) we obtain that \(\mathbf{S}\) is invertible and moreover
Note also from (3.8)
for another constant \({\tilde{C}}_2\).
Let us now return to (3.14) which we write as \(\varvec{\beta }={\mathbf {S}}^{-1} {\mathbf {y}}\). From this we obtain
The above quantity can be made arbitrarily small by a choice of a large K in \(c_1=K \frac{{\mathscr {C}}_2^3}{{\mathscr {C}}_1^3}\). Thus
can be made a small multiple of \({\mathscr {C}}_1\sqrt{\delta }\) by a large choice of K. We have used the upper bound in (3.7) above. Using this along with the lower bound in (3.7) once again we obtain
uniformly in j, and this proves (3.11) if we choose K large enough. We then get the bound in (3.10) with \(\eta =P\left( |N(0,1)|\le \frac{1}{C_3{\mathscr {C}}_1} \right) \) for some constant \(C_3\). \(\square \)
Remark 3.3
One can check that one obtains for small \(\delta \)
In particular if \({\mathscr {C}}_1\) is large one obtains a bound of \(\exp \left( -C_4 \frac{{\mathscr {C}}_1^3\log {\mathscr {C}}_1}{K{\mathscr {C}}_2^3}\right) \).
Remark 3.4
We note that if \(\sigma (s, y)\) is a constant function, then Lemma 3.2 can be easily proved by using Slepian’s inequality. For instance, if \(\sigma (s, y)=1\), then it is easy to see that there exist \(c_0>0\) and \(c_1>0\) such that \(G(t_1, z)\) is convex for all \(|z|\ge c_1\delta \), and then the convexity implies that
Using now Slepian’s inequality, we get
which provides an upper bound on \(P(A_1)\) as in (3.9) since \({\tilde{\Delta }}_k\) is mean-zero Gaussian with variance estimated in (3.7).
3.2 Upper bound in Theorem 1.2 (b)
We provide the outline of the proof of the upper bound. The details are quite similar to that of the proof of the upper bound of Theorem 1.2 (a) and are left to the reader. Using the same discrete time-space points as in (3.1) we obtain
Defining
the upper bound will follow once we show the existence of a \(0<{\tilde{\eta }}<1\) such that
Note here the slight change from (3.5); here we condition on the profile up to \(t_i\). One could have conditioned up to time \(t_{i-1}\) but conditioning up to time \(t_i\) makes the argument simpler. Define
Using (2.19), one obtains a similar result to Lemma 3.1 with the random variables \({\tilde{\Delta }}_k\) replaced by \({\tilde{\Delta }}^{\#}_k\). We also need
By the Markov property again, we only need to show
for some \(0<{\tilde{\eta }}<1\). For this we note
where
Define \(\Delta ^{\#}_k=u(\delta ^2, x_k)-u(0, x_k)\) and then show
for some \(0<{\tilde{\eta }}<1\), where \({\mathcal {G}}_{j-1}\) is the \(\sigma \) algebra generated by the random variables \({\tilde{\Delta }}^{\#}_k,\, k\le j-1\). Note that although
is of a slightly different form than that of (3.12), the term in the square brackets does not play any role in the argument of Lemma 3.2.
3.3 Upper bound in Theorem 1.3 (a)
The function \(\sigma (t,x,u)\) now depends on the third variable, so the resulting random field is no longer Gaussian. Therefore, we will need an alternative argument based on an approximation procedure. For \(\beta >0\), we define the following equation,
Of course, here, we treat \(x\pm \sqrt{\beta t} \in {\mathbf {T}}\).
Existence and uniqueness of a solution to the above equation is not an issue. In fact, this can be easily proved by considering the following Picard iterates:
with \(V^{(\beta ), 0}(t,\,x):=(G_{t}* u_0)(x).\) We will need the following result.
Proposition 3.1
Assume \(\beta t< \frac{1}{4}\). There exist positive constants \(C_1\) and \(C_2\) that are independent of \(\beta \) and t such that
Proof
We use (3.18) and (3.19) to write
For notational convenience, we set \(f(l,t):=\sup _{x\in {\mathbf {T}}}\Vert V^{(\beta ), l}(t,\,x)-V^{(\beta )}(t,\,x)\Vert _p^2\). We now use Burkholder’s inequality and the fact that \(\sigma \) is globally Lipschitz to obtain
where we have used the heat kernel estimate (2.17) to get the last bound in the above (here the value of the constant C changes from line to line, and is independent of \(\beta \) and t). Upon setting \(F(l):=\sup _{s>0}e^{-ks}f(l,s)\), the above immediately yields
Upon choosing \(k=C{\mathscr {D}}^4p^2\) with some large constant C and iterating, we obtain \(F(l)\le C\left( \frac{1}{2}\right) ^l.\) This along with
gives the result. \(\square \)
We also have the following error estimate on the difference between u and \(V^{(\beta )}\).
Proposition 3.2
Assume \(\beta t< \frac{1}{4}\). Then there exist positive constants \(C_1\) and \(C_2\) that are independent of \(\beta \) and t such that
Proof
We use (3.18) and the mild formulation of the u(t, x) to write
We now use Burkholder’s inequality together with the Lipschitz continuity of \(\sigma \) to write
We bound \(I_2\) first. We now use the heat kernel estimate (2.17) and the fact that \(\sigma \) is bounded above to get that
We now set \(F(k):=\sup _{t>0,\,x\in {\mathbf {T}}}e^{-kt}\Vert u(t,x)-V^{(\beta )}(t,\,x)\Vert _p^2\) and bound \(I_1\) as in the Proposition above to obtain
This finishes the proof upon choosing the \(k=C{\mathscr {D}}^4 p^2\) for a large constant C. \(\square \)
We will use the following straightforward consequence of the above:
where \(D_1\) and \(D_2\) are some positive constants. The following lemma along with (3.20) suggests that we can construct independent random variables that are close to u(t, x). The Proof of Lemma 3.3 is essentially the same as that of Lemma 4.4 of [5].
Lemma 3.3
Let \(\beta , t>0\) and \(l\ge 0\). Fix a collection of points \(x_1,x_2,\cdots \in {\mathbf {T}}\) such that the distance between \(x_i\) and \(x_j\) is greater than \(2l\sqrt{\beta t}\) whenever \(i\not =j\). Then \(\{V^{(\beta ),l}(t,\,x_j)\}\) forms a collection of independent random variables.
We can now prove the upper bound. Recall \(\delta =\epsilon ^{\frac{1}{\theta }}\) and the time points \(t_i:=i \delta ^2\) as in (3.1); we have chosen \(c_0=1\). We shall consider now the spatial points \(x_{2j}:=j(\delta +\rho )\) and \(x_{2j-1}:=j(\delta +\rho )-\rho \) for \(j=1, \dots , J\) where \(J:=[1/2(\delta +\rho )]\) and \(\rho := 2|\alpha \log \epsilon |^{\frac{3}{2}}\delta \). Here \(\alpha >4(D_2{\mathscr {D}}^4+1)+16\theta \) is a constant which is independent of \(\epsilon \) and i, j where \(D_2\) is in (3.20). From this definition, we have \(|x_{2j+1} - x_{2j}| = \delta \) and \(|x_{2j+2}-x_{2j+1}| =\rho \) for \(j=0, \dots , J\). As in the proof of the upper bound for the Gaussian case, we have
We will show below that uniformly over initial profiles \(u_0\) (see Remark 3.2)
for some positive constant C. One then uses (3.4) and the Markov property, and notes that the number of time intervals \(I=\left[ \frac{T}{\delta ^2}\right] =\left[ \frac{T}{\epsilon ^{2/\theta }}\right] \) to get the required upper bound.
Let us therefore turn to the proof of (3.21). Using the triangle inequality, the left hand side of (3.21) is bounded above by
Before we consider \(L_1\) and \(L_2\), we define
Let us now consider \(L_1\) first. By Chebyshev’s inequality and (3.20) there exist constants \(C_1>0\) and \(C_2>0\) which are independent of \(\epsilon \) such that
Since \(2J\le 1/(\delta +\rho )\le 1/\delta =\epsilon ^{-1/\theta }\), we have for some other positive constants \({\tilde{C}}_1\) and \({\tilde{C}}_2\) independent of \(\epsilon \)
Let us now consider \(L_2\). First observe that \(W_j:=\left( V^{(\beta ),l}(t_1, x_{2j}), V^{(\beta ),l}(t_1, x_{2j+1}) \right) \) are independent for \(j=0, 1, \dots , J-1\) by Lemma 3.3 since the distance between \(x_{2j+1}\) and \(x_{2j+2}\) is greater than \(2l^{3/2}\sqrt{t_1}\). Thus, we have
Using the triangle inequality, we have
Let us first consider \(L_{22}\). Consider the following martingale \(M_s\) for \(0\le s\le t_1\):
Note that \(M_{t_1}\) is \(u(t_1, x_{2j+1}) - u(t_1, x_{2j})\). The quadratic variation of the martingale is given by
We use (3.7) to obtain
Since \(M_t\) is a continuous martingale, it is a time change of a Brownian motion B, i.e., \(M_t= M_0 + B_{\langle M\rangle _t}\). Hence, recalling \(\delta =\epsilon ^{1/\theta }\), we have
for some \(\gamma <1\) independent of \(\delta \) (but dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2\)). The last inequality can be obtained by a coupling argument as follows: Let \(B^{M_0}\) be a Brownian motion starting at \(M_0\) independent of a standard Brownian motion B starting at 0. Now let X be the process which follows the trajectory of \(B^{M_0}\) till it hits either B or \( -B\), after which it follows the trajectory of B or \(-B \) (depending on which one it hits). Clearly X has the same distribution as \(B^{M_0}\). If \(\inf _{C_0{\mathscr {C}}_1^2\delta \le t\le C_1{\mathscr {C}}_2^2\delta } |X_t |<5\sqrt{\delta }\) then both the events \(\left\{ \inf _{C_0{\mathscr {C}}_1^2\delta \le t\le C_1{\mathscr {C}}_2^2\delta } |B _t|>5\sqrt{\delta }\right\} \) and \(\left\{ \inf _{C_0{\mathscr {C}}_1^2\delta \le t\le C_1{\mathscr {C}}_2^2\delta } |-B _t|>5\sqrt{\delta }\right\} \) cannot occur simulataneously since then \(B^{M_0}\) would have hit B or \(-B\) before entering the strip \([-5\sqrt{\delta }, 5\sqrt{\delta }]\). Thus
Let us now consider \(L_{21}\). Here, (3.24) implies \(L_{21}\) can be made arbitrarily small by choosing \(\epsilon \) small enough. Therefore, there exists a constant \(\eta <1\) independent of \(\epsilon \) such that
which implies from (3.25)
Combining our bounds on \(L_1\) and \(L_2\), we finish the proof. \(\square \)
3.4 Upper bound in Theorem 1.3 (b)
The proof follows a similar strategy to that of the upper bound proved above and we will sketch the proof focusing on the main differences. Note that we use the same choice of \(\beta \) and l as in part (a) (see (3.22)). Then, we have
where the points \(t_i=\delta ^2\) while \(x_j= 4j |\alpha \log \epsilon |^{3/2}\delta ,\; j=0,\,1,\, \ldots , J:= \left[ \frac{1}{c_1|\log \epsilon |\delta }\right] \). Here we choose \(\alpha \) as in part (a) such that \(|x_i-x_j| \ge 2\ell \sqrt{\beta (t_1+\delta ^2)}\). In other words, by our choices of \(x_j\), \(\left\{ V^{(\beta , l)}(t_1+\delta ^2, x_j) -V^{(\beta ),l}(t_1, x_j)\right\} _{j=0}^{J}\) is a collection of independent random variables. Now we have
For the first two terms, we have similar upper bounds as the one given by (3.23);
for some positive constants \(C_1\) and \(C_2\). For the final term, we have
The bound for the last term is similar to the bound given by (3.26). The martingale term is slightly different. For \(0\le s\le t_1+\delta ^2\)
We now use (2.19) and (2.20) to show that there exist constants \(C_3\) and \(C_4\) such that
A similar argument to that of (3.26) shows that
where \(\gamma <1.\) The proof now follows from part (a).
4 Lower bounds
4.1 Lower bound in Theorem 1.2 (a)
Recall our time discretizations from (3.1): \(t_i=ic_0\delta ^2= ic_0 \epsilon ^{\frac{2}{\theta }},\, i=0,1,\ldots , I\), and consider now the events
where the event \(U_i\) puts restriction on the supremum norm of \(u(t,\cdot )\) in the time interval \([t_i, t_{i+1}]\):
and the event \(H_i\) puts restriction on the Hölder norm of u in the time interval \([t_i, t_{i+1}]\):
It is clear that
Similar to the method of the upper bound, our main task will be to obtain a uniform lower bound on \( P\left( B_i \big \vert B_0, B_1\cdots B_{i-1} \right) \). It turns out that with an appropriate choice of \(c_0\) one can in fact obtain such a uniform lower bound. We do this in Lemma 4.3 below (see also Remark 4.1), and then the lower bound in Theorem 1.2 (a) follows immediately. We first need a couple of lemmas which we turn to next.
Lemma 4.1
There exists a constant \({\mathbf {K}}_5\) dependent only on \(\theta \) and \(\alpha _0 = \frac{C(\theta )}{{\mathscr {C}}_2^{2/\theta }}>0\) such that for \(\alpha <\alpha _0\) we have
Proof
We first split \({\mathbf {T}}^2\) into squares S of side length \(\alpha ^{\frac{1}{2}}\epsilon ^{\frac{1}{\theta }}\). By the Gaussian correlation inequality ([10, 16]) we have
For \(k=0,1,\ldots ,\alpha ^{-\frac{1}{2}}\epsilon ^{- \frac{1}{\theta }}-1\), let \(S_k\) be a square in \({\mathbf {T}}^2\) whose center is \(k2^{-\frac{1}{2}}\alpha ^{\frac{1}{2}}\epsilon ^{\frac{1}{\theta }}\) from the diagonal \(x=y\). There are at most \(2\alpha ^{-\frac{1}{2}}\epsilon ^{-\frac{1}{\theta }}\) of such squares. Therefore the above probability is bounded below by
Let us now give a lower bound of the expression inside the square brackets. We first consider the case when \(k\ge 1\). For any \((x,y)\in S_k\) one has a lower bound \(|x-y|\ge \frac{1}{4}\left( k\alpha ^{\frac{1}{\theta }}\epsilon ^{\frac{1}{\theta }}\right) \) and therefore
the last inequality follows from (2.7). Therefore there exists an \(\alpha _1=\frac{C(\theta )}{{\mathscr {C}}_2^{2/\theta }}>0\) small enough such that for all positive \(\alpha < \alpha _1\) one has
Returning to (4.6) we can obtain a lower bound on the product of terms for which \(k\ne 0\) by choosing an \(\alpha _2=\frac{C(\theta )}{{\mathscr {C}}_2^{2/\theta }}>0\) small enough such that for \(\alpha <\alpha _2\) we have
Finally we consider the \(k=0\) term in (4.6). For a small \(\alpha _3=\frac{C(\theta )}{{\mathscr {C}}_2^{2/\theta }}>0\) one has for \(\alpha <\alpha _3 \)
where the first inequality follows by Lemma 2.5. We now use the bounds (4.9) and (4.8) in (4.6). The statement (4.5) follows immediately from this by choosing \(\alpha _0\le \alpha _1\wedge \alpha _2\wedge \alpha _3\) small enough. \(\square \)
Lemma 4.2
There exists a constant \({\mathbf {K}}_6\) dependent only on \(\theta \), and a positive \({\tilde{\alpha }}_0=\frac{C(\theta )}{\max ({\mathscr {C}}_2^4,{\mathscr {C}}_2^{2/\theta })} \) such that for \(\alpha <{\tilde{\alpha }}_0\) small enough one has
Proof
An application of the Gaussian correlation inequality ([10, 16]) gives
We now partition \({\mathbf {T}}\) into disjoint intervals \([a_i, a_{i+1})\) where \(a_i:=i\alpha ^{\frac{1}{2}}\epsilon ^{\frac{1}{\theta }}\) for \(i=1, \dots , {\alpha ^{-\frac{1}{2}}\epsilon ^{-\frac{1}{\theta }}}\). Applying the Gaussian correlation inequality once again and (2.7), one obtains
if \(\alpha < \alpha _4 = \frac{C}{{\mathscr {C}}_2^4}\) is small enough. The result now follows from this and (4.5). \(\square \)
For the next lemma recall the events \(B_i\) defined in (4.1).
Lemma 4.3
For all initial profiles \(u_0\) with \(|u_0(x)|\le \frac{\epsilon ^{\frac{1}{2\theta }}}{3}\) and \({\mathcal {H}}^{(\theta )}(u_0)\le \frac{\epsilon }{3}\), one has
when \( c_0 6^{\frac{2}{\theta }}<{\tilde{\alpha }}_0\), where \({\tilde{\alpha }}_0\) is defined in Lemma 4.2.
Remark 4.1
Note (see Remark 3.2) that the arbitrariness of \(u_0\) assumed above is so that we have the same lower bound for \(P(B_i\vert B_0, B_1,\ldots , B_{i-1})\). This is because given \(B_{i-1}\) the profile \(u(t_{i-1},\cdot )\) has sup norm at most \(\frac{\epsilon ^{\frac{1}{2\theta }}}{3}\) and Hölder norm at most \(\frac{\epsilon }{3}\). One can then use the Markov property and the above result.
Proof
We will use a change of measure argument inspired by a technique in large deviation theory. A similar method was employed in [1]. Consider the measure Q defined by
where
Define
It follows from Lemma 2.1 in [1] that \(\dot{{\widetilde{W}}}\) is a white noise under the measure Q.
By change of measure
and so Cauchy-Schwarz inequality gives
from which we obtain
Now
We next provide a lower bound on \(Q(B_0)\). First observe that
and
The deterministic term in (4.13) is bounded uniformly (in x) by \(\frac{\epsilon ^{\frac{1}{2\theta }}}{3}\) in the interval \([0,t_1]\) and is equal to 0 at the terminal time \(t_1\). Similarly, due to Lemma 2.1, the first term in (4.14) is bounded uniformly (in x, y) by \(\frac{\epsilon }{3}\) in the same interval and is also equal to 0 at the terminal time \(t_1\). We define \(N_1(t, x)\) and \({\widetilde{N}}_1(t, x, y)\) as N(t, x) and \({\widetilde{N}}(t, x, y)\) as in (2.4) and (2.8) respectively but by replacing \(\dot{W}\) by \(\dot{{\widetilde{W}}}\). It therefore follows
as long as \( c_0 6^{\frac{2}{\theta }}<{\tilde{\alpha }}_0\) from Lemma 4.2. If we use (4.15) and (4.12) into (4.11) we obtain
as long as \( c_0 6^{\frac{2}{\theta }}<{\tilde{\alpha }}_0\). \(\square \)
4.2 Lower bound in Theorem 1.2 (b)
The argument in Sect. 4.1 has to be modified at quite a few places. We first note the following lemma which follows immediately from the Gaussian correlation inequality.
Lemma 4.4
There is \(\alpha _0^{\#} = \frac{C(\theta )}{{\mathscr {C}}_2^{2/\theta }}>0\) such that for \(\alpha <\alpha _0^{\#}\) we have
We shall consider time discretizations \(t_i= ic_2\delta ^2=i c_2\epsilon ^{\frac{2}{\theta }},\; i=0,1,\ldots , I\). The constant \(c_2\) will be appropriately chosen so as to get a uniform lower bound on \(P\left( B_i^{\#} \big \vert B_0^{\#}, B_1^{\#}\cdots B_{i-1}^{\#} \right) \) in (4.20) below. It will only depend on \(\theta \) and \({\mathscr {C}}_2\). In this section let
where, similar to Sect. 4.1,
Here, we recall the constant \(\Lambda \) in (1.4). Let us first consider the following lemma.
Lemma 4.5
We have the following inclusion.
Proof
We take a realization \(u(\cdot ,\cdot )\) of the left hand side. We need to show for any \(s<t\in [0,T]\),
Suppose \(s <t\) are both in \([t_i, t_{i+1}]\). Then, since the profile is in \(T_i^{\#}\),
Next we consider the case when \(s\in [0, t_{i-1}]\) and \(t\in [t_i, t_{i+1}]\). In this case we have
since \(u \in \cap _{i=0}^{I-1} U_i^{\#}\). Finally consider \(s\in [t_{i-1}, t_i]\) and \(t\in [t_i, t_{i+1}]\). In this case
This shows that the realization is in \(\left\{ \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}(u)\le \epsilon \right\} \). \(\square \)
Remark 4.2
Observe that the events \(H_i^{\#}\) play no role in the argument above, and we can in fact take the larger set \(\cap _{i=0}^{I} (U_i^{\#}\cap T_i^{\#})\) in the left hand side of (4.19). However, as we will see in Proposition 4.1 below, to get a lower bound on \(P(T_i^{\#})\) we will need a control on the spatial Hölder norms of u as given by the events \(H_i^{\#}\).
From the above lemma
The lower bound follows from the Markov property and the following
Proposition 4.1
Suppose the initial profile \(u_0\) satisfies
Then there exists a constant \({\mathbf {K}}_{10}>0\) dependent only on \(\theta \) and a positive \({\tilde{\alpha }}^{\#}_0= \frac{C(\theta )}{\max ({\mathscr {C}}_2^4,{\mathscr {C}}_2^{2/\theta })}\) such that for all \(c_2<{\tilde{\alpha }}^{\#}_0\) one has
Proof
We work with the measure Q constructed in the Proof of Lemma 4.3. From (4.13) we have
where we recall
Define
For \(s,t\in [0,t_1]\)
The first term on the right is less than \(\frac{\epsilon }{4}\) thanks to Lemma 2.2. The second term is less than \(\frac{\epsilon }{8}\) by the assumption on the initial profile. Now
by the Gaussian correlation inequality. By splitting the interval \({\mathbf {T}}\) into smaller intervals of length \(c_2^{\frac{1}{2}} \epsilon ^{\frac{1}{\theta }}\) and using Gaussian correlation inequality repeatedly (see also Lemma 4.4 and Lemma 4.2) one obtains
as long as \(c_2\) is small enough.
Following the arguments of Lemma 4.3 we obtain
where Q is the measure constructed there. As in (4.12) we have
Using the above along with (4.21), the proof is complete. \(\square \)
4.3 Lower bound in Theorem 1.3 (a)
We begin by describing the idea behind the proof first. The same idea will be used for the Proof of Theorem 1.4. We will consider the following modifications of the temporal discretisation given by (3.1),
Define
and
We consider the event
Our goal is to provide a lower bound on \(P(A_i)\). By the Markov property it is sufficient to obtain a lower bound on \(P(A_0)\) under the assumption that the initial profile satisfies \(|u_0(x)|\le \epsilon ^{\frac{1}{2\theta }}/3\) and \({\mathcal {H}}^{(\theta )}(u_0) \le \epsilon /3\).
Consider the evolution of \(u(t,\cdot )\) in \([0, t_1]\) and write
where \(u_g(t,x)\) solves
with initial profile \(u_0(x)\). Note that the third coordinate in \(\sigma \) is now \(u_0(x)\) and therefore \(u_g\) is a Gaussian random field. Therefore if we define as in (4.1)
with \(U_0^{(g)}\) and \(H_0^{(g)}\) defined similarly as in (4.2) and (4.3) but for the process \(u_g\) in place of u, and with the new value of \(t_1=\delta ^{2+\eta }=\epsilon ^{(2+\eta )/\theta }\).
Now \(B_0^{(g)}\supset \widetilde{B}_0^{(g)}\), where \(\widetilde{B}_0^{(g)}\) is similar to (4.1) but with u replaced by \(u^{(g)}\), \(\epsilon \) replaced by \({\widetilde{\epsilon }}=\frac{\epsilon }{8}\), and \(t_1=c_0({\widetilde{\epsilon }})^{\frac{2}{\theta }}\). Therefore
when \( c_0 6^{\frac{2}{\theta }}<{\tilde{\alpha }}_0\). The difference between u and \(u_g\) is
Consider the set
Define now
and let
Let the event \({\widetilde{V}}\) be the same as V (see (4.24)) but with D replaced by \({\tilde{D}}\). Now
The equality holds because on the event \(A_0\) we have \(\Vert u(t,\cdot ) -u_0\Vert _\infty <2\epsilon ^{\frac{1}{2\theta }}\) (recall that our initial profile is everywhere less than \(\epsilon ^{\frac{1}{2\theta }}/3\)), and so \(D(t,\cdot )={\widetilde{D}}(t,\cdot )\) up to time \(t_1\) on the event \(A_0\). Now we use Remark 2.2 together with the fact that now \(t_1=c_0\epsilon ^{\frac{2}{\theta }+\frac{\eta }{\theta }}\) to obtain
Next we focus on the last term in (4.26). We divide \({\mathbf {T}}^2\) into squares S of side length \(\sqrt{c}_0 \epsilon ^{\frac{1}{\theta }}\). Let
Using Lemma 2.5 (more specifically, Remark 2.3) we obtain
We plug in the bounds (4.28), (4.27) and (4.23) into (4.26) to obtain
The last two terms are much smaller than the first term for small \(\epsilon \). Therefore \(P(A_0) \ge P(B_0^{(g)})/2\) when \(\epsilon \) is small enough. We thus have a lower bound on \(P(A_i)\) for all i. As mentioned earlier, the proof of (1.11) then follows from the Markov property. \(\square \)
4.4 Lower bound in Theorem 1.3 (b)
The argument follows that of Sect. 4.3 with some modifications. Now let \(t_i=ic_2\delta ^{2+\eta \theta }=\epsilon ^{\frac{2}{\theta }+\eta }\). Similar to before, for \(t\in [t_i, t_{i+1}]\), we write \(u(t,x)= u_g^{(i)}(t,x) + D^{(i)} (t,x)\). Here
and
Now define
where \( H_i^{(g),\#},\, T_i^{(g),\#}\) are as in (4.17) and (4.18) but with \(u_g\) in place of u, and with the \(t_i = ic_2\delta ^{2+\eta \theta }=ic_2\epsilon ^{\frac{2}{\theta }+\eta }\). On the other hand we define
Now let
where
It follows from arguments similar to Lemma 4.5 that
where
By the Markov property it is enough to give a lower bound on \(P({\mathscr {B}}_0^{\#})\) under the assumption that \( |u_0(x)|\le \frac{\epsilon ^{\frac{1}{2\theta }+\eta (\frac{1}{4}-\frac{\theta }{2})}}{8c_2^{\frac{\theta }{2}-\frac{1}{4}}}\) and \({\mathcal {H}}_0^{(\theta )}(u) \le \frac{\epsilon }{8\Lambda }\). Let
Let \({\widetilde{V}}_0^{\#}\) be defined as \(V_0^{\#}\) but with \({\widetilde{D}}^{(0)}\) in place of \(D^{(0)}\). Here \({\widetilde{D}}_0\) is as in (4.25) but with the above \(\tau \).
Using Remark 2.2 and the argument in (4.27) we obtain
Similarly using Remark 2.4 we obtain
Lemma 4.6
We have when \(\epsilon \) is small enough
Proof
As in the Proof of Proposition 4.1, with this new choice of \(t_1=c_2\epsilon ^{\frac{2}{\theta }+\eta }\) and with
\({\widetilde{N}}_1^{(g)},\, N_1^{\#, (g)}\) defined in terms of \(N_1^{(g)}\), we have
A lower bound on the last probability is obtained by taking the supremum of \(s\ne t\) over \([0,c_2\epsilon ^{\frac{2}{\theta }}]\) instead of \([0,t_1]\). This gives (when \(c_2\) is chosen small enough)
Next let
Clearly \({\tilde{t}}_1> t_1\) and so a lower bound of the first term on the right of (4.32) is (note the \(t_1\) in the sup has been replaced by \({\tilde{t}}_1\))
when \(c_2<\frac{C(\theta )}{\max ({\mathscr {C}}_2^4, {\mathscr {C}}_2^{2/\theta })}\) is chosen small enough, using the arguments in Lemma 4.2. We have used the Gaussian correlation inequality in the second step. Note that the sup in t in the second probability is over a larger time interval \([0, c_2\epsilon ^{\frac{2}{\theta }}]\).
The event \(B_0^{(g),\#}\) depends on the noise up to time \(c_2\epsilon ^{\frac{2}{\theta }}\), and so
We have the following upper bound (similar to (4.12):
Plugging in the bounds (4.33),(4.34) and (4.36) into (4.35) we obtain the lemma. \(\square \)
From the above lemma as well as (4.30) and (4.31) we obtain
when \(\epsilon \) is small enough, and thus from (4.29) one gets
for some constant \(C(\theta ,{\mathscr {C}}_2)>0\) dependent only on \({\mathscr {C}}_2\) and \(\theta \). This completes the proof of the lower bound since \(\eta \) is arbitrary. \(\square \)
5 Proof of Theorem 1.4
The proofs of the upper bounds in both statements in Theorem 1.4 are the same as that of Theorem 1.3. The proof of the lower bounds follows the same ideas as in the proofs for the lower bounds of Theorem 1.3. We show this only for statement (a); the proof of statement (b) is similar. The only difference as compared to the Proof in Theorem 1.2 (a) is that we revert back to the discretisation given by (3.1). We therefore have
For any fixed \(c_0\), \( {\mathscr {C}}_1\) and \({\mathscr {C}}_2\), we can choose \({\mathscr {D}}\) small enough so that as \(\epsilon \) decreases, the final two term goes to zero much faster than the first term. Therefore for small \(\epsilon \) a lower bound on \(P(A_0)\) (and hence \(P(A_i)\)) is one half times the first term above. An application of the Markov property then finishes the proof. \(\square \)
6 Proof of Theorem 1.1
We first prove the upper bound. This follows immediately from
and Theorem 1.3. Let us turn our attention to the lower bound. In the proof of the lower bound of Theorem 1.3 (b), we let
and redefine
We then have
In addition, similar to (4.30) and (4.31), Remark 2.3 says that
Now it is easy to see that
It then follows quite easily that under the same assumptions of Theorem 1.3, for any \(\eta >0\), there exist positive constants \(C_1,C_2>0\) dependent on \({\mathscr {C}}_1, {\mathscr {C}}_2,\theta ,\eta \) such that
This finishes the Proof of Theorem 1.1. \(\square \)
Remark 6.1
It is easy to see from the argument presented here that under the assumptions of Theorem 1.2 (resp. Theorem 1.4) we have the same bounds as in (1.7) (resp. (1.11)) for the Hölder semi-norm. We leave the verification to the reader.
7 Proofs of Theorems 1.5 and 1.6
The proof of the lower bound of Theorem 1.5 relies heavily on Theorem 2.2 of [9]. We will use some notations from its proof and indicate only the main differences. The proofs of Theorem 1.6 follow from Theorem 1.5 using the same arguments used previously to deal with the non-gaussian case.
Proof of Theorem 1.5 (a)
The upper bound is a result of Lemma 3.2 (recall the event \(A_i\) defined in (3.3)); note that the initial profile in Lemma 3.2 is arbitrary. Indeed, we might condition on the profile at time \(T-c_0\delta ^2\) and conclude from the above lemma that
where \(0<\eta <1\) and \(J=\left[ \frac{1}{c_1\delta }\right] \), and \(c_0=1, c_1, \delta =\epsilon ^{\frac{1}{\theta }}\) are as in Sect. 3.1. From this and (3.4) we obtain
Integrating over the profile \(u(T-c_0\delta ^2,\cdot )\) we obtain the upper bound.
We next turn our attention to the lower bound. As mentioned above, the proof follows along the lines of the proof of Theorem 2.2 of [9], and we just sketch the necessary modifications in the proof. Recall that we assume that our initial profile \(u_0\equiv 0\), and therefore
Defining \(\varvec{\sigma }^2(\gamma ):=E\left[ \left\{ N(T,x+\gamma ) - N(T,x)\right\} ^2\right] \) it follows from the proof of (3.7) that
for \(\gamma >0\) small enough, where C(T) is a constant dependent on T and \(C_1\) is the constant in (3.7). The above is the key ingredient in the proof of the lower bound. We take \(\varvec{\beta }=\theta \) and \(f(x)=x^{\frac{1}{2}-\theta }\) in Theorem 2.2 in [9]. While it is not true that \(\varvec{\sigma }(x)/x^{\varvec{\beta }} f(x)\) is nondecreasing in x as in Theorem 2.2 of [9], a close examination of the proof reveals that all we require is that \(\varvec{\sigma }(ax)/f(ax) \le C_2 a^{\varvec{\beta }} \varvec{\sigma }(x)/ f(x)\) for some positive constant \(C_2\), for all \(0<a<1\) and x small enough. This clearly holds for us. The sequences \(x_l\) and \(y_{j,l}\) encountered in the proof there should be modified by multiplying by \(\frac{1}{C_2}\). Similarly, while going through the arguments of the lower bounds of the terms A, B, C defined in the paper, one just gets an additional constant multiple inside the exponentials and this does not change the result. We leave this routine checking to the interested reader. The lower bound in Theorem 1.5 follows immediately from the lower bound of Theorem 2.2 in [9]. \(\square \)
Proof of Theorem 1.5 (b)
For the upper bound, let \(t_i=i\epsilon ^{\frac{2}{\theta }},\, i=0,1,\cdots I= T\epsilon ^{-\frac{2}{\theta }}\).
By considering the profile at time \(t_i\) we obtain
Note that \({\mathcal {N}}(t_i, t_{i+1}, X)\) is really the noise term from time \(t_i\) to \(t_{i+1}\), that is thinking of time \(t_i\) as the new time zero . Similar to arguments used a few times in this paper we have
which is bounded uniformly (in i) by a number less than 1 (note that the variance of \( {\mathcal {N}}(t_i, t_{i+1}, X)\) is bounded above and below by constant multiples of \(\epsilon ^{\frac{1}{2\theta }}\)). The Markov property then gives the upper bound.
Consider the process \(Y_t:= u(tT,X),\, 0\le t\le 1\). As we are under the assumption \(u_0\equiv 0\) we have
Defining \(\varvec{\sigma }^2(\gamma ):=E\left[ \left( Y_{t+\gamma } - Y_t\right) ^2\right] \) and using (2.19) and (2.20) one obtains
for constants \(C_1, C_2\) independent of T. One can then follow the argument of the lower bound of Theorem 1.5 for the process \(Y_t\), now with \(f(x) = x^{\frac{1}{4}-\frac{\theta }{2}}\) and \(\varvec{\beta }=\frac{\theta }{2}\). \(\square \)
Proof of Theorem 1.6 (a)
The proof of the upper bound is similar to Theorem 1.3 but instead we use (3.21) and note that this bound is uniform over the initial profiles \(u_0\). We can then conclude
Now integrate over the profile at time \(T-c_0\epsilon ^{\frac{2}{\theta }}\).\(\square \)
Proof of Theorem 1.6 (b)
The proof is very similar to that of the proof of the upper bound of Theorem 1.5 (b). The only difference is that now \({\mathcal {N}}(t_i, t_{i+1}, X)\) is no longer Gaussian. For \(t_i\le s\le t_{i+1}\), we note that
is a martingale. Similar arguments to that of the proof of (3.26) and an application of the Markov property complete the proof. \(\square \)
8 Some extensions
In this section, we provide support theorems in the Hölder semi-norm, which are similar to the support theorem in the sup norm in [1]. We provide probabilities that the solution u stays close to a function in Hölder spaces \(C^{\gamma , \beta }\) (see Theorems 8.2 and 8.3 for the precise statements). These theorems are of a different flavour from the support theorem proved in [3], where a description of the support set of the solution is given.
We first consider small ball probabilities of (1.1) with nice drifts. By means of a change of measure argument, we can show that all of our results are still valid when we add a bounded drift term to the equation. Consider
The proof of the following theorem follows exactly the argument given in Section 2.2 of [1] and is left to the reader to verify.
Theorem 8.1
Consider (8.1), where the assumptions (1.2) and (1.3) on \(\sigma \) hold, and \(g(t,x,u):{\mathbf {R}}^+\times {\mathbf {T}}\times {\mathbf {R}}\rightarrow {\mathbf {R}}\) is bounded in absolute value by a constant \({\mathbb {G}}\) and globally Lipschitz in the third variable (that is, there is a \({\tilde{{\mathscr {D}}}}\) such that \(|g(t,x,v)-g(t,x,w)|\le {\tilde{{\mathscr {D}}}} |v-w|\)). Then the statements of Theorems 1.1, 1.2, 1.3 and 1.4 still hold, with the constants now depending additionally on \({\mathbb {G}}\) but not on \({\tilde{{\mathscr {D}}}}\).
Note that the Lipschitz condition on g is just to gaurantee uniqueness and existence of solutions to (8.1). We also have the following result which is analogous to that of Theorem 1.2 of [1].
Proposition 8.1
Consider the solution to (1.1). Let \(h:{\mathbf {R}}^+\times {\mathbf {T}}\rightarrow {\mathbf {R}}\) be a smooth function such that \(h, \partial _t h\) and \(\partial _x^2h\) are uniformly bounded by a constant H. Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\) and suppose that \({\mathcal {H}}_0^{(\theta )}(u-h) \le \frac{\epsilon }{2} \left( 1\wedge \frac{1}{2\Lambda } \right) \) where \(\Lambda \) is given in (1.4).
-
(a)
Suppose that the function \(\sigma (t,x,u)\) is independent of u but satisfies Assumption 1.1. Then there exist positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta \) and H such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }}}\right) \le P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u-h\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}}\right) , \end{aligned}$$and
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }}}\right) \le P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u-h\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}}\right) . \end{aligned}$$ -
(b)
Suppose that \(\sigma (t,x,u)\) is now dependent on u and satisfies both Assumptions 1.1 and 1.2. Then for any \(\eta >0\), there exist positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta \) and H such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2T}{\epsilon ^{{\frac{3}{\theta }}+\eta }}\right) \le P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u-h\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) , \end{aligned}$$and
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }+\eta }}\right) \le P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u-h\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$ -
(c)
Suppose that \(\sigma (t,x,u)\) is again dependent on u and satisfies both Assumptions 1.1 and 1.2. Then there is a \({\mathcal {D}}_0>0\) such that for all \({\mathcal {D}}<{\mathcal {D}}_0\), there exists positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta \) and H such that
$$\begin{aligned} C_1\exp \left( - \frac{C_2T}{\epsilon ^{\frac{3}{\theta }}}\right) \le P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u-h\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) , \end{aligned}$$and
$$\begin{aligned} C_1\exp \left( - \frac{C_2 T}{\epsilon ^{\frac{3}{\theta }}}\right) \le P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u-h\right) \le \epsilon \right) \le C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$
We quickly discuss how the above follows from Theorem 8.1 and the previous resuls. Consider \(w(t,x)=u(t,x)-h(t,x)\). The reader can check that w satisfies
where \(w_0= u_0(x) -h(0,x)\), and \({\tilde{\sigma }}(t,x,w):= \sigma \left( t,x,w+h(t,x)\right) \). One also observes that \({\tilde{\sigma }}\) satisfies the same assumptions as that of \(\sigma \), and the function \(g(t,x):=\frac{1}{2}\partial _x^2 h-\partial _t h\) satisfies the assumptions of Theorem 8.1. Proposition 8.1 follows from Theorem 8.1 applied to w.
We next increase the collection of functions h for which we can prove support theorems similar in spirit to Proposition 8.1.
Definition 8.1
We say that \(f:[0, T]\times {\mathbf {T}}\rightarrow {\mathbf {R}}\) is in \(C^{\gamma , \beta }\) if we have
In other words, \(C^{\gamma , \beta }\) is the set of functions that are uniformly bounded, Hölder continuous with the exponent \(\gamma \) in time and the exponent \(\beta \) in space.
Let \(\psi :{\mathbf {R}}\rightarrow {\mathbf {R}}\) be a non-negative, symmetric and smooth function such that the support of \(\psi \) is in \([-1, 1]\) and \(\int _{{\mathbf {R}}} \psi (x) \, dx=1\). For any positive integer n and \(f\in C^{\gamma , \beta }\), we set \(\psi _n(x):=n \psi (nx)\) and define
where \({\tilde{f}}\) is the periodization of f in the spatial variable x and we also define \({\tilde{f}}(s, x)=f(0, x)\) for \(s< 0\) and \(x\in {\mathbf {R}}\) and \({\tilde{f}}(s, x)=f(T, x)\) for \(s>T\) and \(x\in {\mathbf {R}}\). We have the following bounds on the derivatives of the above function. The proof is straightforward and is therefore omitted.
Lemma 8.1
Suppose \(f\in C^{\gamma , \beta }\). Then there exists a constant \(C>0\) such that for all \(x\in {\mathbf {T}}\) and \(t\in [0, T]\)
The following lemma shows we can approximate \(f\in C^{\gamma , \beta }\) by smooth mollifications of f.
Lemma 8.2
Let \(f:[0,T]\times {\mathbf {T}}\rightarrow {\mathbf {R}}\) be in \(C^{\gamma , \beta }\) for some \(\gamma , \beta \in (0, 1]\). Consider the sequence of smooth functions \(\{f_n\}_{n=1}^{\infty }\) defined by (8.2). Let \(\beta _1 \in (0, \beta )\) and \(\gamma _1 \in (0, \gamma )\). Then, for any fixed \(\epsilon >0\), there exist constants \(C_1(\epsilon )\) and \(C_2(\epsilon )\) such that we have
and
Proof
We start by making the following observation. Since for each \(t\ge 0\), \(f(t, \cdot )\) is Hölder(\(\beta \)) continuous, for \(\beta _1<\beta \) we have
where \(C_1\) is a positive constant that is independent of t. We also have
We therefore obtain
whenever we choose \(|x-y|\le \left( \frac{\epsilon }{2C_1}\right) ^{\frac{1}{\beta -\beta _1}}\). We now consider \(f(t, x) - f_n(t, x)\):
Since \(f\in C^{\gamma , \beta }\), \(\psi _n(x)=0\) if \(|x| > 1/n\) and \(\int \psi _n(x) \, dx =1\), there exists some constant \(C_2>0\) such that
Hence, for all \(t\in [0, T]\) and \(x,y\in {\mathbf {T}}\) satisfying \(|x-y|\ge \left( \frac{\epsilon }{2C_1}\right) ^{\frac{1}{\beta -\beta _1}}\), there exists some constant \(C_3>0\) which only depends on \(\beta , \beta _1\) such that
We have therefore proved (8.3) for all large enough \(n \ge C_1(\epsilon )\) where
For (8.4), we follow the same proof above but switch \(\beta \) by \(\gamma \) to get (8.4). Here, we need \(n\ge C_2(\epsilon )\) where
This completes the proof of the lemma. \(\square \)
Remark 8.1
It is easy to see from (8.3), (8.4) and (8.5) that every \(f\in C^{\gamma , \beta }\) can be approximated by its smooth mollification \(f_n\) in \(\Vert \cdot \Vert _{C^{\gamma _1, \beta _1}}\) for all \(\gamma _1<\gamma \) and \(\beta _1<\beta \). That is, for any \(\epsilon >0\) and for every \(f\in C^{\gamma , \beta }\), there exists a constant \(C(\epsilon )>0\) such that
We can now drop the assumption that h is smooth in Proposition 8.1. We obtain bounds on the small ball probabilities when \(h \in C^{\gamma , \beta }\). The upper bounds remain the same as before, but the lower bounds now depend on \(\beta \) and \(\gamma \). We now have to treat the spatial and temporal regularities of \(u-h\) differently. We first consider the spatial difference of \(u-h\), i.e., \({\mathcal {H}}_t^{(\theta )}(u-h)\).
Theorem 8.2
Consider the solution to (1.1). Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\). Suppose \(h:[0, T]\times {\mathbf {T}}\rightarrow {\mathbf {R}}\) is in \(C^{\gamma ,\beta }\) with \(\frac{1}{2}-\theta <\beta \le 1\) and \(\gamma \in (0,1]\). We also assume \({\mathcal {H}}_0^{(\theta )}(u-h) \le \frac{\epsilon }{4}\). Then we have the following:
-
(a)
Suppose that the function \(\sigma (t,x,u)\) is independent of u but satisfies Assumption 1.1. Then there exist positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta , \beta \) and \(\gamma \) such that
$$\begin{aligned} C_1\exp \left( - C_2 T\left[ \frac{1}{\epsilon ^{\frac{3}{\theta }}}+ \frac{1}{\epsilon ^{\frac{4 (\beta \vee \gamma )}{\gamma (\beta +\theta -1/2)}}} \right] \right)\le & {} P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u-h\right) \le \epsilon \right) \\\le & {} C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}}\right) . \end{aligned}$$ -
(b)
Suppose that \(\sigma (t,x,u)\) is now dependent on u and satisfies both Assumptions 1.1 and 1.2. Then for any \(\eta >0\), there exist positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta , \beta \) and \(\gamma \) such that
$$\begin{aligned} C_1\exp \left( - C_2 T\left[ \frac{1}{\epsilon ^{\frac{3}{\theta }+\eta }}+\frac{1}{\epsilon ^{\frac{4 (\beta \vee \gamma )}{\gamma (\beta +\theta -1/2)}}} \right] \right)\le & {} P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u-h\right) \le \epsilon \right) \\\le & {} C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$ -
(c)
Suppose that \(\sigma (t,x,u)\) is again dependent on u and satisfies both Assumptions 1.1 and 1.2. Then there is a \({\mathcal {D}}_0>0\) such that for all \({\mathcal {D}}<{\mathcal {D}}_0\), there exists positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta , \beta \) and \(\gamma \) such that
$$\begin{aligned} C_1\exp \left( - C_2 T\left[ \frac{1}{\epsilon ^{\frac{3}{\theta }}}+\frac{1}{\epsilon ^{\frac{4 (\beta \vee \gamma )}{\gamma (\beta +\theta -1/2)}}} \right] \right)\le & {} P \left( \sup _{0\le t\le T} {\mathcal {H}}_t^{(\theta )}\left( u-h\right) \le \epsilon \right) \\\le & {} C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$
Proof
The proof is similar to the proof of Theorem 1.2 of [1], but here we use the approximation procedure presented above. That is, we set \(\beta _1:=\frac{1}{2}-\theta \) in Lemma 8.2 and \(n= C_1(\epsilon )\) where \(C_1(\epsilon )\) is given in (8.6), and define a smooth function \(h_n\) by (8.2). Then, we have
Since \({\mathcal {H}}_t^{(\theta )}(u-h) \le {\mathcal {H}}_t^{(\theta )}(u-h_n) + {\mathcal {H}}_t^{(\theta )}(h-h_n)\), (8.9) implies that
By Lemma 8.1, there exists a constant C so that
Here, \(\left( \partial _t - \frac{1}{2}\partial _x^2 \right) h_n(t, x)\) is the drift term when we consider the differential form of \(u(t, x) - h_n(t, x)\). That is, if we let \({\tilde{u}}_n:=u-h_n\), then \({\tilde{u}}\) satisfies
where \({\tilde{\sigma }}_n(t, x, z):=\sigma (t, x, z + h_n(t, x))\). A close inspection of the proof of Theorem 1.2 of [1] shows that in the case of (a)
Here, \(n^4\) comes from (8.10). Recalling the choice of n finishes the proof of the lower bound in part (a). The arguments for the lower bounds in (b) and (c) are similar.
Let us now consider the upper bounds. First, we prove the upper bound in part (a). Here we also use the approximation procedure. That is, we choose and fix n large enough to get (8.9). Then, by triangle inequality, we have
Let \(v_n(t, x):=u(t, x) - h_n(t, x)\). Then, v satisfies
where \(g_n(t, x) = \left( \partial _t - \frac{1}{2}\partial _x^2 \right) h_n (t, x)\). To get the upper bound in part (a), we just follow the proof of the upper bound of Theorem 1.2 (a). Note that the upper bound of Theorem 1.2 (a) is obtained once (3.11) is proved. The only difference from (8.11) to (1.1) is that we have the additional drift term \(g_n(t, x)\) in (8.11). However, the drift term does not have any effect in obtaining the upper bound. More precisely, similar to (3.11), we need to show that there exists a constant \(C>0\) such that
where
and \({\tilde{\Delta }}_j\) is given in (3.6). Here, since \(g_n\) is deterministic, we have
Thus, (3.11) implies (8.12), which leads to the upper bound in part (a), that is the same upper bound in Theorem 1.2.
For the upper bound in part (b), we follow the proof of the upper bound in Theorem 1.3 (a). That is, we add \(-h(t, x)\) to the mild form of u(t, x) and also add \(-h(t, x)\) to \(V^{(\beta )}\) and \(V^{(\beta , l)}\) in (3.18) and (3.19), then follow the proof of the upper bound in Theorem 1.3 (a). It is easy to see that Propositions 3.1 and 3.2, and Lemma 3.3 still hold. In addition, as in (3.26), we can regard \(\left( u(t_1, x_{2j+1}) - u(t_1, x_{2j}) \right) - \left( h(t_1, x_{2j+1}) - h(t_1, x_{2j}) \right) \) as
Following the same proof of (3.26), we obtain
where \(\gamma \) is given in (3.26).
The proof of the upper bound in part (c) is exactly the same as the one for the upper bound in part (b). \(\square \)
Similar to Theorem 8.2, we now provide small ball probabilities of the temporal Hölder semi-norms of \(u-h\). We skip the proof since one can basically follow the Proof of Theorem 8.2.
Theorem 8.3
Consider the solution to (1.1). Let \(0<\theta <\frac{1}{2}\) and \(0<\epsilon <1\). Suppose \(h:[0, T]\times {\mathbf {T}}\rightarrow {\mathbf {R}}\) is in \(C^{\gamma ,\beta }\) with \(\frac{1}{4}-\frac{\theta }{2}<\gamma \le 1\) and \(\beta \in (0,1]\). We also assume \({\mathscr {H}}_0^{(\theta )}(u-h) \le \frac{\epsilon }{4\Lambda }\). Then we have the following:
-
(a)
Suppose that the function \(\sigma (t,x,u)\) is independent of u but satisfies Assumption 1.1. Then there exist positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta , \beta \) and \(\gamma \) such that
$$\begin{aligned} C_1\exp \left( - C_2 T\left[ \frac{1}{\epsilon ^{\frac{3}{\theta }}}+ \frac{1}{\epsilon ^{\frac{4 (\beta \vee \gamma )}{\beta \left( \gamma +\frac{\theta }{2} -\frac{1}{4}\right) }}} \right] \right)\le & {} P \left( \sup _{ x \in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u-h\right) \le \epsilon \right) \\\le & {} C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}}\right) . \end{aligned}$$ -
(b)
Suppose that \(\sigma (t,x,u)\) is now dependent on u and satisfies both Assumptions 1.1 and 1.2. Then for any \(\eta >0\), there exist positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta , \beta \) and \(\gamma \) such that
$$\begin{aligned} C_1\exp \left( - C_2 T\left[ \frac{1}{\epsilon ^{\frac{3}{\theta }+\eta }}+\frac{1}{\epsilon ^{\frac{4 (\beta \vee \gamma )}{\beta \left( \gamma +\frac{\theta }{2} -\frac{1}{4}\right) }}} \right] \right)\le & {} P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u-h\right) \le \epsilon \right) \\\le & {} C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) . \end{aligned}$$ -
(c)
Suppose that \(\sigma (t,x,u)\) is again dependent on u and satisfies both Assumptions 1.1 and 1.2. Then there is a \({\mathcal {D}}_0>0\) such that for all \({\mathcal {D}}<{\mathcal {D}}_0\), there exists positive constants \(C_1, C_2, C_3, C_4>0\) depending on \({\mathscr {C}}_1, {\mathscr {C}}_2, \theta , \beta \) and \(\gamma \) such that
$$\begin{aligned} C_1\exp \left( - C_2 T\left[ \frac{1}{\epsilon ^{\frac{3}{\theta }}}+\frac{1}{\epsilon ^{\frac{4 (\beta \vee \gamma )}{\beta \left( \gamma +\frac{\theta }{2} -\frac{1}{4}\right) }}} \right] \right)\le & {} P \left( \sup _{x\in {\mathbf {T}}} {\mathscr {H}}_x^{(\theta )}\left( u-h\right) \le \epsilon \right) \\\le & {} C_3\exp \left( - \frac{C_4T}{\epsilon ^{\frac{3}{\theta }}|\log \epsilon |^{\frac{3}{2}}}\right) .\end{aligned}$$
We end with a remark.
Remark 8.2
Support theorems involving the Hölder semi-norm used in Theorem 1.1 can be obtained by a combination of Theorems 8.2 and 8.3. We leave these to the reader.
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Acknowledgements
We thank the referees for a careful reading of the paper and for various suggestions which led to improvement in the exposition. We also thank them for the question raised in Remark 1.5. This work was initiated when M.F. and M.J. visited POSTECH in September 2019. They would like to thank the mathematics department for the hospitality during their visit. M.J. was partially supported by SERB MATRICS grant MTR/2020/000453 and a CPDA grant from I.S.I., and K.K. is partially supported by the NRF (National Research Foundation of Korea) grants 2019R1A5A1028324 and 2020R1A2C4002077.
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Foondun, M., Joseph, M. & Kim, K. Small ball probability estimates for the Hölder semi-norm of the stochastic heat equation. Probab. Theory Relat. Fields 185, 553–613 (2023). https://doi.org/10.1007/s00440-022-01153-w
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DOI: https://doi.org/10.1007/s00440-022-01153-w