STRONG UNIQUENESS FOR SDES IN HILBERT SPACES WITH NONREGULAR DRIFT

• 成果类型: Article
• 署名作者: Da Prato, G.; Flandoli, F.; Rockner, M.; Veretennikov, A. Yu.
• 署名单位: Scuola Normale Superiore di Pisa; University of Pisa; University of Bielefeld; University of Leeds; HSE University (National Research University Higher School of Economics); Kharkevich Institute for Information Transmission Problems of the RAS
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1016
• 发表日期: 2016
• 页码: 1985-2023
• 关键词: stochastic-evolution equations; classical dirichlet forms; topological vector-spaces; differentiability

摘要

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.