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

Details Large deviation for diffusions and Hamilton--Jacobi equation in Hilbert spaces Large deviation for diffusions and Hamilton--Jacobi equation in Hilbert spaces

2006-02-28

arxiv.org/abs/math/0602655v1

Annals of Probability 2006, Vol. 34, No. 1, 321-385

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000567 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

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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