Generalized stochastic flow associated to the It\^o SDE with partially Sobolev coefficients and applications

TL;DR

This paper establishes the existence, uniqueness, and stability of generalized stochastic flows for Itô SDEs with partially Sobolev coefficients, and demonstrates their weak differentiability, extending the understanding of stochastic flow regularity.

Contribution

It introduces new results on the well-posedness and stability of stochastic flows for SDEs with partially Sobolev coefficients, including weak differentiability properties.

Findings

Proved existence and uniqueness of generalized stochastic flows.

Established stability under suitable conditions.

Demonstrated weak differentiability of the stochastic flow.

Abstract

We consider the It\^o SDE with partially Sobolev coefficients. Under some suitable conditions, we show the existence, uniqueness and stability of generalized stochastic flows associated to such an equation. As an application, we prove the weak differentiability of the stochastic flow generated by the It\^o SDE with Sobolev coefficients.