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

• 成果类型: Article
• 署名作者: Gassiat, Paul; Gess, Benjamin; Lions, Pierre-Louis; Souganidis, Panagiotis E.
• 署名单位: Universite PSL; Universite Paris-Dauphine; Max Planck Society; University of Bielefeld; Universite PSL; College de France; University of Chicago
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00921-5
• 发表日期: 2020
• 页码: 421-448
• 关键词: scalar conservation-laws; brownian path

摘要

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.