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January 2006 Large deviation for diffusions and Hamilton–Jacobi equation in Hilbert spaces
Jin Feng
Ann. Probab. 34(1): 321-385 (January 2006). DOI: 10.1214/009117905000000567


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.


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Jin Feng. "Large deviation for diffusions and Hamilton–Jacobi equation in Hilbert spaces." Ann. Probab. 34 (1) 321 - 385, January 2006.


Published: January 2006
First available in Project Euclid: 17 February 2006

zbMATH: 1091.60002
MathSciNet: MR2206350
Digital Object Identifier: 10.1214/009117905000000567

Primary: 60F10
Secondary: 49L25 , 60G99 , 60J25

Keywords: large deviation , stochastic evolution equation in Hilbert space , viscosity solution of Hamilton–Jacobi equations

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 1 • January 2006
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