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.
This is a preview of subscription content, access via your institution.







References
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 Subseries
Barron, E.N., Cannarsa, P., Jensen, R., Sinestrari, C.: Regularity of Hamilton–Jacobi equations when forward is backward. Indiana Univ. Math. J. 48(2), 385–409 (1999)
Carmona, R., Delarue, F.: Probabilistic Theory of Mean Field Games with Applications I–II. Springer, New York (2018)
Coghi, M., Gess, B.: Stochastic nonlinear Fokker-Planck equations. Nonlinear Anal. 187, 259–278 (2019)
Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
Es-Sarhir, A., von Renesse, M.-K.: Ergodicity of stochastic curve shortening flow in the plane. SIAM J. Math. Anal. 44(1), 224–244 (2012)
Fehrman, B., Gess, B.: Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise. Arch. Ration. Mech. Anal. 233(1), 249–322 (2019)
Friz, P.K., Gassiat, P., Lions, P.-L., Souganidis, P.E.: Eikonal equations and pathwise solutions to fully non-linear spdes. Stoch. Partial Differ. Equ. Anal. Comput. 5(2), 256–277 (2017)
Gassiat, P.: A stochastic Hamilton–Jacobi equation with infinite speed of propagation. Comptes Rendus Math. Acad. Sci. Paris 355(3), 296–298 (2017)
Gassiat, P., Gess, B.: Regularization by noise for stochastic Hamilton–Jacobi equations. Probab. Theory Relat. Fields 173(3–4), 1063–1098 (2019)
Gess, B., Souganidis, P.E.: Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci. 13(6), 1569–1597 (2015)
Gess, B., Souganidis, P.E.: Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws. Commun. Pure Appl. Math. 70(8), 1562–1597 (2017)
Goldie, C.M., Grübel, R.: Perpetuities with thin tails. Adv. Appl. Probab. 28(2), 463–480 (1996)
Hoel, H., Karlsen, K.H., Risebro, N.H., Storrøsten, E.B.: Path-dependent convex conservation laws. J. Differ. Equ. 265(6), 2708–2744 (2018)
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)
Imhof, J.-P.: A construction of the Brownian path from \({\rm BES}^3\) pieces. Stoch. Process. Appl. 43(2), 345–353 (1992)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. i–le cas stationnaire. Comptes Rendus Math. 343(9), 619–625 (2006)
Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations, Volume 69 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston (1982)
Lions, P.-L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ. Anal. Comput. 1(4), 664–686 (2013)
Lions, P.-L., Souganidis, P.E. (in preparation)
Lions, P.-L., Souganidis, P.E.: Stochastic Viscosity Solutions (in preparation)
Lions, P.-L., Souganidis, P.E.: Stochastic viscosity solutions of spatially dependent Hamilton–Jacobi equations with multiple paths (in preparation)
Lions, P.-L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations. Comptes Rendus Acad. Sci. Paris Sér. I Math. 326(9), 1085–1092 (1998)
Lions, P.-L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. Comptes Rendus Acad. Sci. Paris Sér. I Math. 327(8), 735–741 (1998)
Lions, P.-L., Souganidis, P.E.: Stochastic Viscosity Solutions. Lectures in College de France (2009)
Souganidis, P.E.: Fully nonlinear first- and second-order stochastic partial differential equations. CIME lecture notes, pp. 1–37 (2016)
Vallois, P.: Decomposing the Brownian path via the range process. Stoch. Process. Appl. 55(2), 211–226 (1995)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gassiat, P., Gess, B., Lions, PL. et al. Speed of propagation for Hamilton–Jacobi equations with multiplicative rough time dependence and convex Hamiltonians. Probab. Theory Relat. Fields 176, 421–448 (2020). https://doi.org/10.1007/s00440-019-00921-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-019-00921-5
Keywords
- Stochastic viscosity solutions
- Stochastic Hamilton–Jacobi equations
- Speed of propagation
Mathematics Subject Classification
- 60H15
- 35D40