Journal
PROBABILITY THEORY AND RELATED FIELDS
Volume 176, Issue 1-2, Pages 421-448Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00440-019-00921-5
Keywords
Stochastic viscosity solutions; Stochastic Hamilton-Jacobi equations; Speed of propagation
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Funding
- ANR [ANR-16-CE40-0020-01]
- Office for Naval Research [N000141712095]
- DFG [CRC 1283]
- National Science Foundation [DMS-1600129]
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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.
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