The Annals of Probability

Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift

G. Da Prato, F. Flandoli, E. Priola, and M. Röckner

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We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov’s fundamental result on $\mathbb{R}^{d}$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin–Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.

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Ann. Probab., Volume 41, Number 5 (2013), 3306-3344.

First available in Project Euclid: 12 September 2013

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Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60H15: Stochastic partial differential equations [See also 35R60]

Pathwise uniqueness stochastic PDEs bounded measurable drift


Da Prato, G.; Flandoli, F.; Priola, E.; Röckner, M. Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 (2013), no. 5, 3306--3344. doi:10.1214/12-AOP763.

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