[Paper Review] Semilinear stochastic partial differential equations: central limit theorem and moderate deviations

Rangrang Zhang, Jie Xiong|arXiv (Cornell University)|Mar 30, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance14 references3 citations

TL;DR

This paper establishes the central limit theorem (CLT) and moderate deviation principles (MDP) for a class of semilinear stochastic partial differential equations (SPDEs) driven by space-time white noise on a bounded domain. Using a weak convergence approach and variational representations, it proves that the normalized deviation of the solution from its deterministic limit converges to a Gaussian process under CLT scaling and satisfies an MDP under intermediate scaling, applicable to equations like stochastic Burgers and reaction-diffusion types.

ABSTRACT

In this paper, we establish a central limit theorem (CLT) and the moderate deviation principles (MDP) for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to stochastic partial differential equations of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.

Motivation & Objective

• To establish the central limit theorem (CLT) for the solution of a semilinear SPDE driven by space-time white noise.
• To derive the moderate deviation principle (MDP) for the same class of SPDEs, bridging the gap between CLT and large deviations.
• To address the challenge of estimating the supremum of stochastic integrals with time-dependent, non-semimartingale integrands in the context of space-time white noise.
• To extend existing results on CLT and MDP from time-white noise models to the more complex space-time white noise setting.
• To provide a rigorous foundation for statistical inference via asymptotic confidence intervals in SPDE models with multiplicative noise.

Proposed method

• Applies the weak convergence approach of Dupuis and Ellis to prove the MDP, relying on a variational representation formula for the Laplace transform of bounded continuous functionals.
• Uses the Boué-Dupuis and Budhiraja-Dupuis variational representations to link the Laplace principle to the MDP for SPDEs with space-time white noise.
• Analyzes the deviation process $ X^\varepsilon(t) = \frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}(u^\varepsilon - u^0)(t) $ under intermediate scaling $ \lambda(\varepsilon) \to \infty $, $ \sqrt{\varepsilon}\lambda(\varepsilon) \to 0 $.
• Employs a perturbation argument via the stochastic integral representation of the solution difference, decomposing the deviation into a limiting Gaussian process and a remainder term.
• Establishes tightness and convergence in law of the normalized deviation process by controlling the supremum norm of stochastic integrals using Itô's formula and Gronwall-type estimates.
• Proves that the limiting process satisfies a linearized SPDE driven by the noise coefficient $ \sigma $, with drift terms from the derivatives of $ b $ and $ g $, under the assumption of linear growth and quadratic growth conditions.

Experimental results

Research questions

• RQ1 Does the solution of a semilinear SPDE with space-time white noise satisfy a central limit theorem as the noise intensity $ \varepsilon \to 0 $?
• RQ2 Can a moderate deviation principle be established for such SPDEs when the scaling $ \lambda(\varepsilon) $ lies between the CLT and large deviation regimes?
• RQ3 How can one control the supremum of a stochastic integral with a time-dependent, non-semimartingale integrand in the context of space-time white noise?
• RQ4 What is the limiting behavior of the normalized deviation $ X^\varepsilon $ in $ C([0,T];L^2([0,1])) $ under intermediate scaling?
• RQ5 Does the limiting process in the MDP have the same law as the solution of a linearized SPDE driven by the noise coefficient $ \sigma $?

Key findings

• The central limit theorem holds for the solution of the semilinear SPDE with space-time white noise, where the normalized deviation $ X^\varepsilon $ converges in law to a Gaussian process in $ C([0,T];L^2([0,1])) $.
• The moderate deviation principle is established for the SPDE under intermediate scaling $ \lambda(\varepsilon) \to \infty $, $ \sqrt{\varepsilon}\lambda(\varepsilon) \to 0 $, with the rate function derived via the weak convergence approach.
• The limiting process in the MDP satisfies a linearized SPDE driven by the noise coefficient $ \sigma $, with additional drift terms arising from the Fréchet derivatives of $ b $ and $ g $.
• The proof overcomes the technical challenge of estimating the supremum of stochastic integrals with time-dependent integrands by using a decomposition into a limiting Gaussian process and a remainder that vanishes in probability.
• The convergence in law of the normalized solution $ \mathcal{G}^\varepsilon(W + \lambda(\varepsilon)\int h^\varepsilon) $ to $ \mathcal{G}^0(\int h) $ is established, confirming the MDP under the given scaling.
• The results apply to important models such as the stochastic Burgers equation and stochastic reaction-diffusion equations with space-time white noise, extending prior results on LDP and CLT to the MDP regime.