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

• 成果类型: Article
• 署名作者: Gordina, Maria; Roeckner, Michael; Teplyaev, Alexander
• 署名单位: University of Connecticut; University of Bielefeld; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00991-w
• 发表日期: 2020
• 页码: 861-891
• 关键词: invariant-measures; quasi-invariance; dirichlet forms; EQUATIONS

摘要

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.