seminar: Regularization of ordinary and partial differential equations by noise

speaker: Michael Röckner (Faculty of Mathematics, Bielefeld University)

abstract: It is a well-known phenomenon that an ordinary differential equation becomes ``more regular'', if one adds a noise term, as e.g. a stochastic differential given by a Brownian motion. On the level of the associated Fokker-Planck-Kolmogorov equations (FPKE), whose solutions are just the transition probabilities of the resulting solution process, this becomes more or less obvious, since the FPKE becomes elliptic, if the noise is not degenerate. From a purely analytic point of view, this regularizing property of the noise is most impressively manifested by the fact that noise can ``produce'' (existence and, in particular) uniqueness of solutions . Indeed, e.g. a classical result of A. Yu.Veretennikov (see 1 and the references therein), tells us that, given an initial condition, any two corresponding solutions of an ordinary differential equation in d-dimensional Euclidean space given by a just measurable bounded vector field and perturbed by the differential of a d-dimensional Brownian path, coincide for almost every such path. In contrast to this, in the deterministic case, neither existence nor uniqueness of solutions hold in such a case. The purpose of this talk is to present recent results of the same type, but for partial differential equations perturbed by noise, i.e. for the infinite dimensional analogue of the situation described above. 1 N.V. Krylov, M. Roeckner, Strong solutions of stochastic equations with singular time-dependent drift, Probab. Theory Rel. Fields 131 (2005), no. 2, 154-196. 2 G. Da Prato, F. Flandoli, E. Priola, M. Roeckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces with bounded measurable drift, preprint, 2011.

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