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Session 34. SPDE: stochastic analysis and dynamics
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Strong uniqueness for SDEs in Hilbert spaces with non-regular drift |
Prof. Dr. Michael Röckner, Bielefeld University, Germany
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The talk is based on the joint work with Giuseppe Da Prato, Franco Flandoli and Alexander Veretennikov.
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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 subdifferential 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 non-degenerate 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.
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