Functional Inequalities for Non-Symmetric Stochastic Differential Equations

TL;DR

This paper investigates conditions under which non-symmetric stochastic differential equations exhibit functional inequalities similar to symmetric cases, ensuring properties like exponential convergence and hypercontractivity.

Contribution

It establishes criteria for non-symmetric SDEs to satisfy functional inequalities, extending symmetric case results to non-symmetric Markov processes.

Findings

Non-symmetric Markov semigroups can satisfy functional inequalities under certain conditions.

Conditions are identified for SDEs driven by Brownian motion or Lévy jumps.

Results show non-symmetric processes can share properties like hypercontractivity with symmetric ones.

Abstract

According to the theory of functional inequalities, a non-symmetric Markov semigroup has better properties than the corresponding symmetric one. For instance, there exist 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 aim to search for reasonable conditions to ensure that a non-symmetric Markov semigroup and its symmetrization share the properties of exponential convergence, uniform integrability, hypercontractivity, and supercontractivity. 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. SDEs driven by Brownian motion or L\'evy jump process are investigated.