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Moderate Deviations for a Stochastic Heat Equation with Spatially Correlated Noise

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Abstract

In this paper, we proved a central limit theorem and established a moderate deviation principle for a perturbed stochastic heat equations defined on [0,T]×[0,1]d. This equation is driven by a Gaussian noise, white in time and correlated in space.

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Acknowledgements

The authors are grateful to the anonymous referees for conscientious comments and corrections. Y. Li and S. Zhang were supported by Natural Science Foundation of China (11471304, 11401556). R. Wang was supported by Natural Science Foundation of China (11301498, 11431014).

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Correspondence to Ran Wang.

Appendix

Appendix

To make reading easier, we present here some results on the kernel G partially from Márquez-Carreras and Sarrà [26].

Recall that G denotes the fundamental solution of the heat equation

$$ \begin{cases} \frac{\partial}{\partial t}G_t(x,y)=\Delta_x G_t(x,y),& t\ge0,\ x,y\in[0,1]^d,\\ G_t(x,y)=0,& x\in\partial([0,1]^d),\\ G_0(x,y)=\delta(x-y). \end{cases} $$
(53)

The details about the construction of the fundamental solution G can be found in [18, Chap. 1]. This fundamental solution G is non-negative and can be decomposed into different terms as follows (see [15]):

$$ G_{t-s}(x,y)=H(t-s,x-y)+R(t-s,x,y)+\sum _{i\in I_d}H_i(t-s,x-y), $$
(54)

where H is the heat kernel on \(\mathbb{R}^{d}\),

$$H(t,x)= \biggl(\frac{1}{4\pi t} \biggr)^{\frac{d}{2}}\exp \biggl(- \frac {|x-y|^2}{4t} \biggr), $$

R is a Lipschitz function, I d ={i=(i 1,…,i d )∈{−1,0,1}d−{(0,…,0)}} and

$$H_i(t-s,x-y)=(-1)^kH\bigl(t-s,x-y^i \bigr), $$

with \(k=\sum_{j=1}^{d}|i_{j}|\) and

$$ \begin{cases} y_j^i=y_j, & \mbox{if}\ i_j=0, \\ y_j^i=-y_j, & \mbox{if}\ i_j=1, \\ y_j^i=2-y_j, & \mbox{if}\ i_j=-1. \\ \end{cases} $$

Lemma 5.1

Assume (H η ) for some η∈(0,1). There exists a positive constant C independent of t and x such that, for any \(0\le t'< t\le T, x\in[0,1]^{d},\gamma _{1}\in(0,\frac{1-\eta}{2})\), \(1\le p<1+\frac{2}{d}\) and \(\gamma _{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\),

$$\begin{aligned} &\int_0^{t'}\bigl\Vert G_{t-s}(x,\cdot)-G_{t'-s}(x,\cdot )\bigr\Vert _{\mathcal{H}}^2ds\le C|t-t'|^{2\gamma_1}, \end{aligned}$$
(55)
$$\begin{aligned} & \int_{t'}^t\bigl\Vert G_{t-s}(x,\cdot)\bigr\Vert _{\mathcal{H}}^2ds\le C|t-t'|^{2\gamma_1}, \end{aligned}$$
(56)
$$\begin{aligned} & \int_{0}^{t'}ds\int _{[0,1]^d}dy|G_{t-s}(x,y)-G_{t'-s}(x,y)|^p \le C|t-t'|^{2\gamma_2}, \end{aligned}$$
(57)
$$\begin{aligned} & \int_{t'}^tds\int _{[0,1]^d}dy |G_{t-s}(x,y)|^p\le C|t-t'|^{2\gamma_2}. \end{aligned}$$
(58)

Proof

The proof of (55) and (56) can be found in Lemma 6.1.3 in [26]. The proofs of (57) and (58) are similar. For the convenience of reading, we shall give the proofs of (57) and (58).

For any \(p\in[1,1+\frac{2}{d})\), (54) implies that

$$\begin{aligned} |G_{t-s}(x,y)-G_{t'-s}(x,y)|^p \le&C(d, p) \biggl(\big|H(t-s,x-y)-H\bigl(t'-s,x-y\bigr)\big|^p \\ &{}+ \big|R(t-s,x-y)-R\bigl(t'-s,x-y\bigr)\big|^p \\ &{}+\sum_{i\in I_d}\big|H_i(t-s,x-y)-H_i \bigl(t'-s,x-y\bigr)\big|^p \biggr). \end{aligned}$$
(59)

In order to check (57), we only need to bound the right-hand side of (59). Let L be the Lipschitz constant of the function R. Clearly

$$ \int_{0}^{t'}ds\int _{[0,1]^d}dy\big|R(t-s,x,y)-R\bigl(t'-s, x,y \bigr)\big|^p\le L^pT|t-t'|^{p}. $$
(60)

The terms H i can be studied using the same arguments as H. Now it remains to analyze the term with H. First, for any \(p\in[1,1+\frac{2}{d})\),

$$ \int_{0}^{t'}ds\int _{[0,1]^d}dy\big|H(t-s,x-y)-H\bigl(t'-s,x-y \bigr)\big|^p\le C (A_1+A_2), $$
(61)

with

$$\begin{aligned} &A_1=\int_0^{t'}ds\int _{[0,1]^d}dy \biggl(\frac{1}{4\pi(t-s)} \biggr)^{\frac{pd}{2}} \biggl\vert \exp \biggl(-\frac{|x-y|^2}{4(t-s)} \biggr)-\exp \biggl(- \frac {|x-y|^2}{4(t'-s)} \biggr)\biggr\vert ^p, \\ &A_2=\int_0^{t'}ds\int _{[0,1]^d}dy\biggl\vert \biggl(\frac{1}{4\pi (t-s)} \biggr)^{\frac{d}{2}}- \biggl(\frac{1}{4\pi(t'-s)} \biggr)^{\frac{d}{2}}\biggr\vert ^{p} \exp \biggl(-\frac{p|x-y|^2}{4(t'-s)} \biggr). \end{aligned}$$

Since t′<t, we have

$$\exp \biggl(-\frac{|x-y|^2}{4(t-s)} \biggr)\ge\exp \biggl(-\frac {|x-y|^2}{4(t'-s)} \biggr). $$

Using the inequality

$$(a-b)^p\le a^p-b^p \quad\mbox{for all } a>b>0,\ p\ge1, $$

we obtain that

$$\begin{aligned} A_1&\le\int_0^{t'}ds\int _{\mathbb{R}^d}dy \biggl(\frac{1}{4\pi (t-s)} \biggr)^{\frac{pd}{2}} \biggl[\exp \biggl(-\frac {p|x-y|^2}{4(t-s)} \biggr)-\exp \biggl(-\frac{p|x-y|^2}{4(t'-s)} \biggr) \biggr] \\ &=\int_0^{t'} \biggl(\frac{1}{4\pi(t-s)} \biggr)^{\frac {d(p-1)}{2}}p^{-\frac{d}{2}} \biggl[1- \biggl(\frac{t'-s}{t-s} \biggr)^{\frac{d}{2}} \biggr]. \end{aligned}$$

Since for any \(\gamma_{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\) and s<t′,

$$\frac{t-t'}{t-s}< \biggl(\frac{t-t'}{t-s} \biggr)^{2\gamma_2}\le1,\quad\quad \biggl(\frac{t'-s}{t-s} \biggr)^{n}\le\frac{t'-s}{t-s}<1,\quad n\ge1, $$

we have

$$\begin{aligned} 1- \biggl(\frac{t'-s}{t-s} \biggr)^{\frac{d}{2}} &\le \biggl[1- \biggl( \frac{t'-s}{t-s} \biggr) \biggr]\times \biggl(1+\frac{t'-s}{t-s}+ \biggl( \frac{t'-s}{t-s} \biggr)^2+\cdots+ \biggl(\frac{t'-s}{t-s} \biggr)^{ \lfloor\frac{d}{2} \rfloor } \biggr) \\ &\le \biggl( \biggl\lfloor \frac{d}{2} \biggr\rfloor +1 \biggr) \biggl( \frac{t-t'}{t-s} \biggr) \\ &\le \biggl( \biggl\lfloor \frac{d}{2} \biggr\rfloor +1 \biggr) \biggl( \frac{t-t'}{t-s} \biggr)^{2\gamma_2}, \end{aligned}$$

where \(\lfloor\frac{d}{2} \rfloor\) is the largest integer number less or equal than \(\frac{d}{2}\).

Thus, for any \(\gamma_{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\), we have

$$\begin{aligned} A_1&\le (4\pi)^{-\frac{d(p-1)}{2}}p^{-\frac{d}{2}} \biggl( \biggl\lfloor \frac{d}{2} \biggr\rfloor +1 \biggr)\cdot|t-t'|^{2\gamma_2} \cdot\int_0^{t'}ds\bigl(t'-s \bigr)^{-\frac{d(p-1)}{2}-2\gamma_2} \\ &\le C(p,\gamma_2,d, T)|t-t'|^{2\gamma_2}. \end{aligned}$$
(62)

Similar computations to the study of A 2, we have

$$\begin{aligned} A_2&\le\int_0^{t'}ds \int_{\mathbb{R}^d}dy\biggl\vert \biggl(\frac {1}{4\pi(t-s)} \biggr)^{\frac{d}{2}}- \biggl(\frac{1}{4\pi (t'-s)} \biggr)^{\frac{d}{2}}\biggr\vert ^{p} \exp \biggl(-\frac{p|x-y|^2}{4(t'-s)} \biggr) \\ &=\int_0^{t'}ds\biggl\vert \biggl( \frac{1}{4\pi(t-s)} \biggr)^{\frac {d}{2}}- \biggl(\frac{1}{4\pi(t'-s)} \biggr)^{\frac{d}{2}}\biggr\vert ^{p}\cdot \biggl(\frac{4\pi(t'-s)}{p} \biggr)^{\frac{d}{2}} \\ &=(4\pi)^{-\frac{d(p-1)}{2}}p^{-\frac{d}{2}}\int_0^{t'}ds|(t-s)^{-\frac{d}{2}}- \bigl(t'-s\bigr)^{-\frac{d}{2}}|^{p-1}\cdot \biggl(1- \biggl(\frac {t'-s}{t-s} \biggr)^{\frac{d}{2}} \biggr) \\ &\le(4\pi)^{-\frac{d(p-1)}{2}}p^{-\frac{d}{2}}\int_0^{t'}ds \bigl(t'-s\bigr)^{-\frac{d(p-1)}{2}}\cdot \biggl(1- \biggl( \frac {t'-s}{t-s} \biggr)^{\frac{d}{2}} \biggr) \\ &\le(4\pi)^{-\frac{d(p-1)}{2}}p^{-\frac{d}{2}} \biggl( \biggl\lfloor \frac{d}{2} \biggr\rfloor +1 \biggr) \cdot|t-t'|^{2\gamma_2} \cdot\int_0^{t'}ds\bigl(t'-s \bigr)^{-\frac{d(p-1)}{2}-2\gamma_2} \\ &\le C(p,d,\gamma_2,T)|t-t'|^{2\gamma_2}. \end{aligned}$$
(63)

Then (57) follows form (59)–(63).

Next we shall prove (58). As before, we only need to check (58) replacing G by H. For any 0≤t′≤tT, \(1\le p<1+\frac{2}{d}\) and \(\gamma_{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\), we have

$$\begin{aligned} &\int_{t'}^tds\int_{[0,1]^d}dyH(t-s,x-y)^p \\ &\quad\le\int_{t'}^tds\int_{\mathbb{R}^d}dy \biggl(\frac{1}{4\pi (t-s)} \biggr)^{\frac{pd}{2}}\exp \biggl(-\frac {p|x-y|^2}{4(t-s)} \biggr) \\ &\quad=(4\pi)^{-\frac{(p-1)d}{2}}p^{-\frac{d}{2}}\int_{t'}^tds(t-s)^{-\frac {(p-1)d}{2}} \\ &\quad=(4\pi)^{-\frac{(p-1)d}{2}}p^{-\frac{d}{2}} \biggl(1-\frac {(p-1)d}{2} \biggr)^{-1}\bigl(t-t'\bigr)^{1-\frac{(p-1)d}{2}} \\ &\quad\le C(p,\gamma_2, d, T)|t-t'|^{2\gamma_2}. \end{aligned}$$

The proof is complete. □

Lemma 5.2

Assume (H η ) for some η∈(0,1). There exists a positive constant C independent of t and x such that, for any 0≤tT,x,x′∈[0,1]d, \(p\in [1,1+\frac{2}{d})\), γ 1∈(0,1−η) and \(\gamma_{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\),

$$\begin{aligned} & \int_0^{t}\bigl\Vert G_{t-s}(x,\cdot)-G_{t-s}\bigl(x',\cdot\bigr)\bigr\Vert _{\mathcal{H}}^2ds\le C|x-x'|^{2\gamma_1}, \end{aligned}$$
(64)
$$\begin{aligned} & \int_{0}^{t}ds\int _{[0,1]^d}dy\big|G_{t-s}(x,y)-G_{t-s} \bigl(x',y\bigr)\big|^p\le C|x-x'|^{2\gamma_2}. \end{aligned}$$
(65)

Proof

The proof of (64) can be found in Lemma 6.1.4 in [26]. Now we shall give the proof of (65). Using the similar arguments as in the proof of (57), we only need to check (65) replacing G by H. For any x,x′∈[0,1]d, \(p\in[1,1+\frac{2}{d})\) and \(\gamma _{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\),

$$\begin{aligned} B :=& \int_{0}^{t}ds\int_{[0,1]^d}dy\big|H(t-s,x-y)-H \bigl(t-s, x'-y\bigr)\big|^p \\ \le& C\int_{0}^{t}ds\int_{\mathbb{R}^d}dy \bigl(4\pi(t-s) \bigr)^{-\frac{pd}{2}}\biggl\vert \exp \biggl(- \frac{|x-y|^2}{4(t-s)} \biggr)-\exp \biggl(-\frac{|x'-y|^2}{4(t-s)} \biggr)\biggr\vert ^{2\gamma_2} \\ &{}\times \biggl(\exp \biggl(-\frac{(p-2\gamma_2)|x-y|^2}{4(t-s)} \biggr)+\exp \biggl(- \frac{(p-2\gamma_2)|x'-y|^2}{4(t-s)} \biggr) \biggr). \end{aligned}$$

By the mean-value theorem, and the fact e x<0 for any x>0, we have

$$\begin{aligned} B \le& C\int_{0}^{t}ds\int_{\mathbb{R}^d}dy \bigl(4\pi(t-s) \bigr)^{-\frac{pd}{2}} \biggl(\frac{|x-x'|}{t-s} \biggr)^{2\gamma_2} \\ &{}\times \biggl(\exp \biggl(-\frac{(p-2\gamma_2)|x-y|^2}{4(t-s)} \biggr)+\exp \biggl(- \frac{(p-2\gamma_2)|x'-y|^2}{4(t-s)} \biggr) \biggr) \\ \le& 2C(4\pi)^{-\frac{d(p-1)}{2}}(p-2\gamma_2)^{-\frac{d}{2}}\cdot |x-x'|^{2\gamma_2}\cdot\int_0^tds(t-s)^{-\frac{d(p-1)}{2}-2\gamma _2} \\ \le& C(p,\gamma_2,d,T)|x-x'|^{2\gamma_2}. \end{aligned}$$

The proof is complete. □

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Li, Y., Wang, R. & Zhang, S. Moderate Deviations for a Stochastic Heat Equation with Spatially Correlated Noise. Acta Appl Math 139, 59–80 (2015). https://doi.org/10.1007/s10440-014-9969-x

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