Central Limit Theorem and Moderate Deviations for a Class of Semilinear Stochastic Partial Differential Equations
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
CENTRAL LIMIT THEOREM AND MODERATE DEVIATIONS FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS∗
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Shulan HU (
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China E-mail :hu [email protected]
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Ruinan LI (
School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai 201620, China E-mail :[email protected]
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Xinyu WANG (
†
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China E-mail :wang xin [email protected] Abstract In this paper we prove a central limit theorem and a moderate deviation principle for a class of semilinear stochastic partial differential equations, which contain the stochastic Burgers’ equation and the stochastic reaction-diffusion equation. The weak convergence method plays an important role. Key words
stochastic Burgers’ equation; stochastic reaction-diffusion equation; large deviations; moderate deviations
2010 MR Subject Classification
1
60H15; 60F05; 60F10
Introduction
For any ε > 0, consider the semilinear stochastic partial differential equation (SPDE for short) √ ∂U ε ∂2U ε ∂2W (t, x) = (t, x) + εσ (t, x, U ε (t, x)) (t, x) 2 ∂t ∂x ∂t∂x ∂ + g (t, x, U ε (t, x)) + f (t, x, U ε (t, x)) ∂x ∗ Received
(1.1)
October 14, 2018; revised October 28, 2019. The research of HU was supported by NSFF (17BTJ034). The research of WANG was supported by NSFC (11871382, 11771161). † Corresponding author: Xinyu WANG.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
for all (t, x) ∈ [0, T ] × [0, 1], with Dirichlet boundary conditions (U ε (t, 0) = U ε (t, 1) = 0) and initial condition U ε (0, x) = η(x) ∈ Lp ([0, 1]), p ≥ 2; W denotes the Brownian sheet defined on a probability space (Ω, F , {Ft }t≥0 , P); the coefficients f = f (t, x, r), g = g(t, x, r), σ = σ(t, x, r) are Borel functions of (t, x, r) ∈ R+ × [0, 1] × R (see Section 2 for details). This family of semilinear equations contain both the stochastic Burgers’ equation and the stochastic reactiondiffusion equation; see Gy¨ongy [15] for details. Intuitively, as the parameter ε tends to zero, the solutions U ε of (1.1) will tend to the solution of ∂U 0 ∂2U 0 ∂ (t, x) = (t, x) + g t, x, U 0 (t, x) + f t, x, U 0 (t, x) (1.2) ∂t ∂x2 ∂x for all (t, x) ∈ [0, T ] × [0, 1], with the Dirichlet boundary conditions and initial condition η(x). It is always interesting to investigate deviations of U ε from the deterministic solution U 0 as ε decreases to 0; that is, the asymptotic behavior of the trajectory 1 U ε − U 0 (t, x), (t, x) ∈ [0, T ] × [0, 1], X ε (t, x) := √ ελ(ε)
where λ(ε) is some deviation scale which strongly influences the asymptotic behavior of X ε . √ (1) The case λ(ε) = 1/ ε provides some large deviation estimates. Cardon-Weber [6] studied the large deviations for the small noise limit of stochastic semilinear SPDEs by the exponential approximations. Ve
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