Ornstein-Uhlenbeck processes with singular drifts: integral estimates and Girsanov densities

TL;DR

This paper introduces pseudo-weak solutions for Hilbert space-valued Ornstein-Uhlenbeck processes with singular, time-dependent drifts, establishing their existence, pathwise estimates, and absolute continuity of laws, along with integrability of Girsanov densities.

Contribution

It develops a new framework for solutions to singular drift equations, proving existence, pathwise estimates, and law absolute continuity, extending results to non-autonomous and finite-dimensional cases.

Findings

Pseudo-weak solutions always exist and have continuous paths.

Laws of solutions are absolutely continuous with respect to original process.

Girsanov densities associated with solutions are integrable.

Abstract

We consider a perturbation of a Hilbert space-valued Ornstein--Uhlenbeck process by a class of singular nonlinear non-autonomous maximal monotone time-dependent drifts. The only further assumption on the drift is that it is bounded on balls in the Hilbert space uniformly in time. First we introduce a new notion of generalized solutions for such equations which we call pseudo-weak solutions and prove that they always exist and obtain pathwise estimates in terms of the data of the equation. Then we prove that their laws are absolutely continuous with respect to the law of the original Ornstein--Uhlenbeck process. In particular, we show that pseudo-weak solutions always have continuous sample paths. In addition, we obtain integrability estimates of the associated Girsanov densities. Some of our results concern non-random equations as well, while probabilistic results are new even in finite-dimensional autonomous settings.