Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians
- Authors: Paul Gassiat, Benjamin Gess, Pierre-Louis Lions, Panagiotis E. Souganidis
- Preprint year: 2018
- First public date: 2018-05-22
- arXiv: 1805.08477
- Status: Published
- Publication type: Journal article
- Publication year: 2020
- Journal: Probability theory and related fields, 176 (2020) 1/2, pp. 421-448
- DOI: 10.1007/s00440-019-00921-5
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.