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Large deviation for diffusions and Hamilton-Jacobi equation in Hilbert spaces
Feng, Jin
Feng, Jin
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Abstract
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-spacevalued
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.
Description
This is the published version, also available here: http://dx.doi.org/10.1214/009117905000000567.
Date
2006-01-01
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Institute of Mathematical Statistics
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FengJin_AP_34(1)321.pdf
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Keywords
large deviation, stochastic evolution equation in Hilbert space, viscosity solution of Hamilton-Jacobi equations
Citation
Feng, Jin. "Large deviation for diffusions and Hamilton-Jacobi equation in Hilbert spaces." The Annals of Probability. (2006) 34, 1. 321-385. http://dx.doi.org/10.1214/009117905000000567.