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# Communications in Mathematical Sciences

## Volume 9 (2011)

### Number 1

### A singular 1-D Hamilton–Jacobi equation, with application to large deviation of diffusions

Pages: 289 – 300

DOI: http://dx.doi.org/10.4310/CMS.2011.v9.n1.a14

#### Authors

#### Abstract

The comparison principle (uniqueness) for the Hamilton-Jacobi equation is usually established through arguments involving a distance function. In this article we illustrate the subtle nature of choosing such a distance function, using a special example of one dimensional Hamiltonian with coefficient singularly (non-Lipschitz) depending upon the state variable. The standard method of using Euclidean distance as a test function fails in such situation. Once the comparison is established, we apply it to obtain a new result on small noise Freidlin-Wentzell type probabilistic large deviation theorem for certain singular diffusion processes.

This article serves to explain basic ideas behind an abstract approach to comparison developed in [J. Feng and T.G. Kurtz, American Mathematical Society, Providence, Rhode Island. Mathematical Surveys and Monographs, 131, 2006], [J. Feng and M. Katsoulakis, Arch. Ration. Mech. Anal., 192(2), 275-310, 2009] in a simple manner, removing all technicalities due to infinite dimensionality.

#### Keywords

comparison principle, singular Hamiltonian, large deviation for singular diffusions

#### 2010 Mathematics Subject Classification

49L25, 60F10