Harnack Inequalities for Stochastic (Functional) Differential Equations with Non-Lipschitzian Coefficients

TL;DR

This paper establishes Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients using coupling methods, providing new results on solution existence and uniqueness.

Contribution

It introduces novel coupling techniques to derive Harnack inequalities for equations with non-Lipschitz coefficients, expanding the theoretical understanding of such stochastic systems.

Findings

Harnack inequalities are proved for a broad class of stochastic differential equations.

New existence and uniqueness results for solutions on open domains are provided.

The methods handle multiplicative noise and non-Lipschitz conditions effectively.

Abstract

By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two results on existence and uniqueness of solutions on an open domain are presented.