The Annals of Probability

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

G. Da Prato, F. Flandoli, M. Röckner, and A. Yu. Veretennikov

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Abstract

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada–Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 1985-2023.

Dates
Received: April 2014
Revised: January 2015
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1463410037

Digital Object Identifier
doi:10.1214/15-AOP1016

Mathematical Reviews number (MathSciNet)
MR3502599

Zentralblatt MATH identifier
1347.60077

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 31C25: Dirichlet spaces 60J25: Continuous-time Markov processes on general state spaces

Keywords
Pathwise uniqueness stochastic differential equations on Hilbert spaces stochastic PDEs maximal regularity on infinite dimensional spaces (classical) Dirichlet forms exceptional sets

Citation

Da Prato, G.; Flandoli, F.; Röckner, M.; Veretennikov, A. Yu. Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. Ann. Probab. 44 (2016), no. 3, 1985--2023. doi:10.1214/15-AOP1016. https://projecteuclid.org/euclid.aop/1463410037


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