On the speed of propagation for stochastic Hamilton-Jacobi equations

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Paul Gassiat
Université Paris Dauphine

Abstract: We study the speed of propagation of initial data for Hamilton-Jacobi equations with multiplicative rough (typically stochastic) time dependence. We first show that, in contrast with the classical (deterministic) case, in general this speed may be infinite as soon as the driving noise has unbounded variation. In the case where the Hamiltonian is convex in the gradient, we show that the range of dependence is bounded by a multiple of the length of a piecewise linear path obtained by connecting the successive extrema of the original path. When the driving path is a Brownian motion, this implies finite speed of propagation. Based on a joint work with B. Gess, P. Souganidis and P.L. Lions.