[Paper Review] Large deviation for two-time-scale stochastic Burgers equation

Xiaobin Sun, Ran Wang+2 more|arXiv (Cornell University)|Nov 1, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance29 references3 citations

TL;DR

This paper establishes a Freidlin-Wentzell type large deviation principle (LDP) for a two-time-scale stochastic Burgers equation with a slow component governed by a stochastic Burgers equation and a fast component by a stochastic reaction-diffusion equation. Using the weak convergence criterion from Budhiraja and Dupuis (2010), the authors prove the LDP by analyzing the asymptotic behavior of a controlled system under small noise and fast-slow time-scale separation.

ABSTRACT

A Freidlin-Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [3].

Motivation & Objective

• To establish a Freidlin-Wentzell type large deviation principle (LDP) for a coupled slow-fast system of stochastic partial differential equations (SPDEs), where the slow component is a stochastic Burgers equation and the fast component is a stochastic reaction-diffusion equation.
• To analyze the rare event behavior of the slow component in the presence of small noise and fast-scale dynamics, under the framework of two-time-scale stochastic systems.
• To extend existing LDP results for multiscale diffusions to infinite-dimensional SPDEs with nonlinear and non-gradient structures.
• To simplify the analysis of the LDP by applying the weak convergence criterion from Budhiraja and Dupuis (2010), avoiding more complex viscosity or weak convergence methods.

Proposed method

• Applies the weak convergence criterion from Budhiraja and Dupuis (2010) to prove the LDP for the slow component of the two-time-scale SPDE system.
• Introduces a controlled version of the original system, where the noise is replaced by a control process $ u^ ho $, enabling the use of the weak convergence approach.
• Analyzes the convergence of the controlled process $ (X^{ ho, ho,u^ ho}, Y^{ ho, ho,u^ ho}) $ as $ ho o 0 $, under the time-scale separation $ ho = ho( ho) $, with $ ho $ small.
• Uses the convergence of the controlled process in distribution to the limit governed by the averaged equation, under appropriate compactness and continuity conditions.
• Employs Sobolev embedding and regularity estimates for the semigroup $ e^{tA} $, and bounds on the nonlinear bilinear operator $ b(x,y,z) $, to control the nonlinearities in the Burgers and reaction-diffusion equations.
• Establishes tightness and compactness of the image of the control map under the weak convergence criterion, ensuring the existence of a rate function.

Experimental results

Research questions

• RQ1 Does a Freidlin-Wentzell type large deviation principle hold for a two-time-scale system where the slow component is a stochastic Burgers equation and the fast component is a stochastic reaction-diffusion equation?
• RQ2 Can the weak convergence criterion from Budhiraja and Dupuis (2010) be effectively applied to prove the LDP for such a multiscale SPDE system?
• RQ3 What is the structure of the rate function for the large deviation principle in this two-time-scale SPDE context?
• RQ4 How do the nonlinear terms in the Burgers equation and the coupling between slow and fast components affect the large deviation behavior?
• RQ5 What are the necessary regularity and growth conditions on the coefficients $ f, g, ho_1, ho_2 $ to ensure the validity of the LDP?

Key findings

• A Freidlin-Wentzell type large deviation principle is established for the slow component $ X^{ ho, ho} $ of the two-time-scale SPDE system, with the rate function defined via the weak convergence criterion.
• The rate function $ I(g) $ is given by $ \frac{1}{2} \int_0^T \|u(s)\|^2 ds $, where $ u $ is a control process such that $ g = \Gamma^0(\int_0^\cdot u(s) ds) $, with $ \Gamma^0 $ being the limit of the controlled system.
• The convergence of the controlled process $ (X^{ ho, ho,u^ ho}, Y^{ ho, ho,u^ ho}) $ in distribution to a deterministic limit is proven under the weak convergence criterion, under appropriate conditions on the coefficients and noise operators.
• The analysis relies on uniform estimates in Sobolev norms for the semigroup $ e^{tA} $, including bounds of the form $ \|e^{tA}x\|_\gamma \leq C t^{-(\gamma-\theta)/2} \|x\|_\theta $, which control the regularity of solutions.
• The bilinear operator $ b(x,y,z) $ associated with the Burgers nonlinearity satisfies $ |b(x,y,z)| \leq C \|x\|_{\alpha_1} \|y\|_{\alpha_2+1} \nolinebreak \|z\|_{\alpha_3} $ under suitable index conditions, ensuring boundedness and continuity.
• The compactness of the image of the control map in the state space is established, which is essential for the weak convergence method to apply and for the LDP to hold.