Analytic theory of It\^o-stochastic differential equations with non-smooth coefficients

TL;DR

This paper provides a comprehensive analysis of Itô stochastic differential equations with non-smooth coefficients, establishing conditions for key properties like existence, uniqueness, and long-term behavior under minimal regularity assumptions.

Contribution

It introduces explicit conditions for fundamental properties of SDEs with low regularity coefficients, extending classical results to more general, non-smooth cases.

Findings

Established non-explosion and irreducibility under minimal regularity

Derived sharp conditions for existence and uniqueness of solutions

Analyzed long-term behavior and invariant measures

Abstract

We present a detailed analysis of non-degenerate time-homogeneous It\^o-stochastic differential equations with low local regularity assumptions on the coefficients. In particular the drift coefficient may only satisfy a local integrability condition. We discuss non-explosion, irreducibility, Krylov type estimates, regularity of the transition function and resolvent, moment inequalities, recurrence, transience, long time behavior of the transition function, existence and uniqueness of invariant measures, as well as pathwise uniqueness, strong solutions and uniqueness in law. This analysis shows in particular that sharp explicit conditions for the various mentioned properties can be derived similarly to the case of classical stochastic differential equations with local Lipschitz coefficients.