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

Xiaoxue Deng (Department of Mathematical Sciences, Tsinghua University, Beijing)

Jin Feng (Department of Mathematics, University of Kansas)

Yong Liu (LMAM, School of Mathematical Sciences, Institute of Mathematics, Peking University, Beijing)

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

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