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 $\R^d$ 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.

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

PRIOLA, Enrico;
2013-01-01

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 $\R^d$ 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.
2013
41
5
3306
3344
http://arxiv.org/pdf/1109.0363v3
http://www.imstat.org/aop/
Pathwise uniqueness; stochastic PDEs; bounded measurable drift
G. Da Prato; F. Flandoli; E. Priola; M. Rockner
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/122673
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