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

  • Paul Gassiat
  • Benjamin Gess
  • Pierre-Louis LionsEmail author
  • Panagiotis E. Souganidis
Article

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.

Keywords

Stochastic viscosity solutions Stochastic Hamilton–Jacobi equations Speed of propagation 

Mathematics Subject Classification

60H15 35D40 

Notes

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.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Paul Gassiat
    • 1
  • Benjamin Gess
    • 2
    • 3
  • Pierre-Louis Lions
    • 1
    • 4
    Email author
  • Panagiotis E. Souganidis
    • 5
  1. 1.CEREMADEUniversité Paris-Dauphine, PSL UniversityParis Cedex 16France
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.Faculty of MathematicsUniversity of BielefeldBielefeldGermany
  4. 4.Collège de FranceParisFrance
  5. 5.Department of MathematicsUniversity of ChicagoChicagoUSA

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