A norm equivalence result for stochastic differential equations with locally Lipschitz coefficients

TL;DR

This paper proves a norm equivalence for SDEs with locally Lipschitz coefficients and derives an SDE for the inverse flow under weaker regularity assumptions, advancing theoretical understanding of stochastic flows.

Contribution

It introduces a two-sided weighted integrability estimate for SDEs with locally Lipschitz coefficients and derives an inverse flow SDE under relaxed regularity conditions.

Findings

Established a norm equivalence for SDEs with locally Lipschitz coefficients.

Derived an SDE for the inverse stochastic flow under weaker assumptions.

Provided new tools for analyzing stochastic flows with less regular coefficients.

Abstract

We establish two-sided weighted integrability estimates, often referred to as a norm equivalence result, for stochastic differential equations (SDEs) with locally Lipschitz coefficients. As a key ingredient in our approach, we also derive an SDE satisfied by the inverse stochastic flow under reduced regularity assumptions in the globally Lipschitz setting.