# Stochastic variational inequalities and applications to the total variation flow pertubed by linear multiplicative noise

Wednesday, October 24, 2012 - 2:00pm - 2:50pm

Keller 3-180

Michael Röckner (Universität Bielefeld)

This is joint work with Viorel Barbu (Romanian Academy, Iasi).

We extend the approach of variational inequalities (VI) to partial differential equations (PDE) with singular coefficients, to the stochastic case. As a model case we concentrate on the parabolic 1-Laplace equation (a PDE with highly singular diffusivity) on a bounded convex domain in N-dimensional Euclidean space, perturbed by linear multiplicative noise, where the latter is given by a

function valued (infinite dimensional) Wiener process. We prove existence and uniqueness of solutions for the corresponding stochastic variational inequality (SVI) in all space dimensions N and for any square-integrable initial condition, thus obtaining a stochastic version of the (minimal) total variation flow. One main tool to achieve this, is to transform the SVI and its approximating stochastic PDE into a deterministic VI, PDE respectively, with random coefficients, thus gaining sharper spatial regularity results for the solutions. We also prove finite time extinction of solutions with positive probability in up to N = 3 space dimensions.

We extend the approach of variational inequalities (VI) to partial differential equations (PDE) with singular coefficients, to the stochastic case. As a model case we concentrate on the parabolic 1-Laplace equation (a PDE with highly singular diffusivity) on a bounded convex domain in N-dimensional Euclidean space, perturbed by linear multiplicative noise, where the latter is given by a

function valued (infinite dimensional) Wiener process. We prove existence and uniqueness of solutions for the corresponding stochastic variational inequality (SVI) in all space dimensions N and for any square-integrable initial condition, thus obtaining a stochastic version of the (minimal) total variation flow. One main tool to achieve this, is to transform the SVI and its approximating stochastic PDE into a deterministic VI, PDE respectively, with random coefficients, thus gaining sharper spatial regularity results for the solutions. We also prove finite time extinction of solutions with positive probability in up to N = 3 space dimensions.

MSC Code:

49K20

Keywords: