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

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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.

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Acknowledgements

The authors thank G. Da Prato, M. Hairer, M. Hinz and N. Krylov for helpful discussions and suggestions. The authors are grateful to anonymous referees for corrections and suggested improvements to the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Maria Gordina & Alexander Teplyaev

  2. Department of Mathematics, Bielefeld University, 33501, Bielefeld, Germany

    Michael Röckner

  3. Academy of Mathematics and Systems Science, CAS, Beijing, China

    Michael Röckner

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  1. Maria Gordina
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Correspondence to Maria Gordina.

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M. Gordina: Research was supported in part by NSF Grant DMS-1712427, and a Simons Fellowship. M. Röckner: Research was supported in part by the German Science Foundation (DFG) through CRC 1283. A. Teplyaev: Research was supported in part by NSF Grant DMS-1613025.

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Gordina, M., Röckner, M. & Teplyaev, A. Ornstein–Uhlenbeck processes with singular drifts: integral estimates and Girsanov densities. Probab. Theory Relat. Fields 178, 861–891 (2020). https://doi.org/10.1007/s00440-020-00991-w

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  • DOI: https://doi.org/10.1007/s00440-020-00991-w

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