Speed of propagation for Hamilton–Jacobi equations with multiplicative rough time dependence and convex Hamiltonians
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.
KeywordsStochastic viscosity solutions Stochastic Hamilton–Jacobi equations Speed of propagation
Mathematics Subject Classification60H15 35D40
Gassiat was partially supported by the ANR via the project ANR-16-CE40-0020-01. Souganidis was partially supported by the National Science Foundation Grant DMS-1600129 and the Office for Naval Research Grant N000141712095. Gess was partially supported by the DFG through CRC 1283.
- 1.Bardi, M., Crandall, M.G., Evans, L.C., Soner, H.M., Souganidis, P.E.: Viscosity Solutions and Applications, Volume 1660 of Lecture Notes in Mathematics. Springer, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence (1997). Lectures given at the 2nd C.I.M.E. Session held in Montecatini Terme, June 12–20, 1995, Edited by I. Capuzzo Dolcetta and P. L. Lions, Fondazione CIME/CIME Foundation SubseriesGoogle Scholar
- 15.Hoel, H., Karlsen, K.H., Risebro, N.H., Storrøsten, E.B.: Numerical methods for conservation laws with rough flux. arXiv preprint arXiv:1802.00708 (2018)
- 18.Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations, Volume 69 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston (1982)Google Scholar
- 20.Lions, P.-L., Souganidis, P.E. (in preparation) Google Scholar
- 21.Lions, P.-L., Souganidis, P.E.: Stochastic Viscosity Solutions (in preparation) Google Scholar
- 22.Lions, P.-L., Souganidis, P.E.: Stochastic viscosity solutions of spatially dependent Hamilton–Jacobi equations with multiple paths (in preparation) Google Scholar
- 25.Lions, P.-L., Souganidis, P.E.: Stochastic Viscosity Solutions. Lectures in College de France (2009)Google Scholar
- 26.Souganidis, P.E.: Fully nonlinear first- and second-order stochastic partial differential equations. CIME lecture notes, pp. 1–37 (2016)Google Scholar