Large deviation for diffusions and Hamilton-Jacobi equation in hilbert spaces

• 成果类型: Article
• 署名作者: Feng, J
• 署名单位: University of Massachusetts System; University of Massachusetts Amherst
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000567
• 发表日期: 2006
• 页码: 321-385
• 关键词: markov process expectations; partial-differential equations; unbounded linear terms; VISCOSITY SOLUTIONS; infinite dimensions; asymptotic evaluation; large time; vanishing viscosity; perturbations; PRINCIPLE

摘要

Large deviation for Markov processes can be studied by Hamilton-Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton-Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) holds for the limit equation. Finally, we verify an exponential compact containment estimate. The large deviation principle then follows from the above three verifications. This paper illustrates such a method applied to a class of Hilbert-space-valued small diffusion processes. The examples include stochastically perturbed Allen-Cahn, Cahn-Hilliard PDEs and a one-dimensional quasilinear PDE with a viscosity term. We prove the comparison principle using a variant of the Tataru method. We also discuss different notions of viscosity solution in infinite dimensions in such context.