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

By Giuseppe Da Prato, Franco Flandoli, Enrico Priola, Michael Röckner. 2011

129 citations recorded by the source corpus.

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 Rd 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.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/22da9a6a29f652d5288a5c77e1a3c67ca248453c

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Citations in the corpus

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift (2014, methodology): “This result was first proved in [11] in the case V = 0 (see also the more recent [12], where also the case V = 0, but with B only bounded on balls was treated) with a rather complex proof based on the very nontrivial maximal regularity results in L(H,γ ) for the Kolmogorov equation (λ −LA,B)u = f as”
Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift (2014, methodology): “Comparing with the above mentioned results of [5, 6, 7], Theorem 1.”
Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients (2014, methodology): “ere an interesting open question is that whether we can extend the above characterization to the inﬁnite dimensional case in somehow so tha t it can be used to the SDE in Hilbert spaces as studied in [7, 8, 9, 10]. Now, we turn to the study of SDE (1.1) and make the following assumptions on σ and b.”
Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation (2012, methodology): “We also refer to the recent article [5] for an extension of the previous results to a Hilbert space setting.”