Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians

2018/05/22 by Paul Gassiat, Benjamin Gess, Gassiat, Paul +5
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.1805.08477

arxiv created 2019/06/25 · arxiv updated 2019/06/26

Abstract

We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.

Related

• A stochastic Hamilton-Jacobi equation with infinite speed of propagation
• New regularity results and long time behavior of pathwise (stochastic) Hamilton-Jacobi equations
• Homogenization of Hamilton-Jacobi equations with rough time dependence
• Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations
• Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton-Jacobi equations