Applications of Stochastic Partial Differential Equations

s of the talks Robert Adler, Technion-Israel Institute of Technology, Israel On quantifying shape, with two applications to stochastic processes I shall discuss some classical Integral and Differential Geometric ways to classify shape, and describe 1. A new class of results about the excursion sets of smooth random fields which uses them. 2. An application of these classifiers to the study of the motion of manifolds under random flows. Rainer Buckdahn, Université de Bretagne Occidentale. France Stochastic viscosity solution for SPDEs driven by a Brownian motion The talk is based on two recent common works with J.Ma/I.Bulla and J.Ma/N.Mrhardy, respectively, and will present a general approach for stochastic viscosity solution for nonlinear stochastic partial differential equations (SPDEs) du(t, x) = F (x, (u,Dxu,D 2 xu)(t, x))dt+ g(x, (u,Dxu)(t, x)) ◦ dWt, t ∈ [0, T ], u(0, x) = u0(x), x ∈ R, driven by a Brownian motion W ; the stochastic integral is interpreted in the sense of Stratonovich. It generalizes former works by P.L.Lions and P.E.Souganidis as well as by J.Ma and R.B. investigating stochastic viscosity solution separately in two particular cases of the above equation: While in the case where g is independent of Dxu a Doss-Sussman type transformation was applied in order to transform the SPDE into a pathwise PDE, the case of a u-independent diffusion term g was studied with the help of the method of stochastic characteristics. However, a generalization of former results by J.Ma and R.B. about the stochastic Taylor expansion allow to generalize the method of stochastic characteristics and to introduce the notion of stochastic viscosity solution for SPDEs in which the diffusion coefficient can depend on both u and its gradient Dxu. The talk will be completed by a uniqueness result for stochastic viscosity solutions.