A stochastic Hamilton-Jacobi equation with infinite speed of propagation

TL;DR

This paper presents an example of a stochastic Hamilton-Jacobi equation demonstrating infinite speed of propagation when the driving signal is not of bounded variation, highlighting a fundamental property of such equations.

Contribution

It introduces a specific stochastic Hamilton-Jacobi equation with infinite propagation speed under certain conditions, revealing new insights into its behavior.

Findings

Infinite speed of propagation occurs when the driving signal lacks bounded variation.

The example clarifies the impact of the driving signal's regularity on the equation's dynamics.

Highlights a key difference from classical Hamilton-Jacobi equations with bounded variation signals.

Abstract

We give an example of a stochastic Hamilton-Jacobi equation $du = H(Du) d\xi$ which has an infinite speed of propagation as soon as the driving signal $\xi$ is not of bounded variation.