Analytic theory of Itô-stochastic differential equations with non-smooth coefficients

Haesung Lee, Wilhelm Stannat, Gerald Trutnau

Published 2020-12-28Version 1

We present a detailed analysis of 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 conditions can in this situation be derived similarly to the case of classical stochastic differential equations with local Lipschitz coefficients and closes hereby a gap in the existing literature.