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

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.