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

TL;DR

This paper demonstrates that Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians exhibit finite speed of propagation, with the dependence range controlled by the path's skeleton, especially for Brownian motion.

Contribution

It establishes finite speed of propagation for such equations and links the dependence range to the skeleton of the driving path, including Brownian motion.

Findings

Dependence range bounded by the skeleton length of the path.

Brownian motion's skeleton has almost surely finite length.

Discussion on the optimality of the estimate.

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.