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May 2016 Strong uniqueness for SDEs in Hilbert spaces with nonregular drift
G. Da Prato, F. Flandoli, M. Röckner, A. Yu. Veretennikov
Ann. Probab. 44(3): 1985-2023 (May 2016). DOI: 10.1214/15-AOP1016

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.

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G. Da Prato. F. Flandoli. M. Röckner. A. Yu. Veretennikov. "Strong uniqueness for SDEs in Hilbert spaces with nonregular drift." Ann. Probab. 44 (3) 1985 - 2023, May 2016. https://doi.org/10.1214/15-AOP1016

Information

Received: 1 April 2014; Revised: 1 January 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1347.60077
MathSciNet: MR3502599
Digital Object Identifier: 10.1214/15-AOP1016

Subjects:
Primary: 31C25, 35R60, 60H15, 60J25

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • May 2016
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