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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991841

Digital Object Identifier
doi:10.1214/12-AOP763

Mathematical Reviews number (MathSciNet)
MR3127884

Zentralblatt MATH identifier
1291.35455

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

Keywords
Pathwise uniqueness stochastic PDEs bounded measurable drift

Citation

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. https://projecteuclid.org/euclid.aop/1378991841


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