Functional analytic approach to stochastic differential equations I

Michael Roeckner (University of Bielefeld)

Friday 13th June, 2014 10:00-11:00 Maths 516


In the fi rst lecture a new general approach to solve stochastic partial di erential equations (SPDE)
will be presented. This new approach is based on a so-called \rescaling transformation", which
transforms the SPDE into a (deterministic) PDE with random coecients (RPDE). This RPDE
can then be considered as a nonlinear operatorial equation on an appropiate Geldfand triple consisting of Lp-spaces of maps depending on time and ! 2, where , equipped with -algebra
F and a probability measure P, is the underlying probability space. To this operatorial equation
standard nonlinear functional analytic methods apply, since for a large class of SPDE the corresponding operatorial equation satis es classical monotonicity conditions. In particular, if the drift of the initial SPDE is the subdi erential (or gradient) of a convex functional, then the solution
of the SPDE is identi ed as the solution of a minimization problem. Applications include porous
media, p-Laplace and transport equations perturbed by linear multiplicative noise.

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