Stochastic scalar conservation laws

In this course we will consider stochastic scalar conservation laws of the type $$du + \sum_{i=1}^d \partial_{x_i} A^i(x,u)\circ d\beta^i_t =0,$$ 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 $$du + \sum_{i=1}^d \partial_{x_i} A^i(u)\circ d\beta^i_t =0,\qquad (\star)$$ 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