Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

Da Prato G, Flandoli F, Röckner M, Veretennikov AY (2016)
The Annals of Probability 44(3): 1985-2023.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose non-linear 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.
Erscheinungsjahr
Zeitschriftentitel
The Annals of Probability
Band
44
Ausgabe
3
Seite(n)
1985-2023
ISSN
PUB-ID

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Da Prato G, Flandoli F, Röckner M, Veretennikov AY. Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. The Annals of Probability. 2016;44(3):1985-2023.
Da Prato, G., Flandoli, F., Röckner, M., & Veretennikov, A. Y. (2016). Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. The Annals of Probability, 44(3), 1985-2023. doi:10.1214/15-AOP1016
Da Prato, G., Flandoli, F., Röckner, M., and Veretennikov, A. Y. (2016). Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. The Annals of Probability 44, 1985-2023.
Da Prato, G., et al., 2016. Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. The Annals of Probability, 44(3), p 1985-2023.
G. Da Prato, et al., “Strong uniqueness for SDEs in Hilbert spaces with nonregular drift”, The Annals of Probability, vol. 44, 2016, pp. 1985-2023.
Da Prato, G., Flandoli, F., Röckner, M., Veretennikov, A.Y.: Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. The Annals of Probability. 44, 1985-2023 (2016).
Da Prato, Giuseppe, Flandoli, F., Röckner, Michael, and Veretennikov, A. Yu. “Strong uniqueness for SDEs in Hilbert spaces with nonregular drift”. The Annals of Probability 44.3 (2016): 1985-2023.