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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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
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]):
where H is the heat kernel on \(\mathbb{R}^{d}\),
R is a Lipschitz function, I d ={i=(i 1,…,i d )∈{−1,0,1}d−{(0,…,0)}} and
with \(k=\sum_{j=1}^{d}|i_{j}|\) and
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 \(\gamma _{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\),
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
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
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 \(p\in[1,1+\frac{2}{d})\),
with
Since t′<t, we have
Using the inequality
we obtain that
Since for any \(\gamma_{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\) and s<t′,
we have
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
Similar computations to the study of A 2, we have
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′≤t≤T, \(1\le p<1+\frac{2}{d}\) and \(\gamma_{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\), we have
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≤t≤T,x,x′∈[0,1]d, \(p\in [1,1+\frac{2}{d})\), γ 1∈(0,1−η) and \(\gamma_{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\),
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 \(\gamma _{2}\in[0,\frac{1}{2}-\frac{d(p-1)}{4})\),
By the mean-value theorem, and the fact e −x<0 for any x>0, we have
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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DOI: https://doi.org/10.1007/s10440-014-9969-x