Stochastic scalar conservation laws

In this course we will consider stochastic scalar conservation laws of the type $$du+\sum _{i=1}^d{\partial }_{x_i}{A}^i(x,u)\circ d{\beta }_t^i=0,$$ for fluxes $A$ being regular enough and ${\beta }^i$ being independent Brownian motions.
These equations appear as continuum limits of mean-field interacting particle systems subject to a common noise and as a simplified part of the mean field game system with common noise.
We will first introduce a notion of an entropy solution to these equations and prove their well-posedness. In the case of spatially homogeneous fluxes $$du+\sum _{i=1}^d{\partial }_{x_i}{A}^i(u)\circ d{\beta }_t^i=0,(\star )$$ we will analyze the regularity of solutions by means of averaging Lemmata. In certain cases we will see that the regularity of solutions to $(\star )$ is higher than in the deterministic case.

Date and time info
Thursday 11:15 - 12:45

Keywords
Stochastic Partial Differential Equations, Stochastic Analysis

Prerequisites
basic measure theory, functional analysis and probability theory

Audience
MSc students, PhD students, Postdocs

Language
English