Functional inequalities for non-symmetric stochastic differential equations

Abstract

A non-symmetric Markov semigroup usually has better properties than the corresponding symmetric one.For example, Wang (2017) provides a class of non-symmetric Markov semigroups which are hypercontractive (and thus converge exponentially in both $L^2$ and entropy),but the symmetric ones are even not ergodic. In this paper, we consider the inverse problem: search for reasonable conditions to ensure that a non-symmetric Markov semigroupand its symmetrization share the properties of exponential convergence, uniform integrability, hypercontractivity, and super boundedness.Since in the symmetric case these properties are precisely characterized by functional inequalities of the Dirichlet form,the key point of the study is to prove these inequalities for non-symmetric Markov processes. Stochastic differential equations driven by Brown motion or Levy jump process are investigated.