A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.’s
Open Access
- 14 April 2008
- journal article
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 360 (9) , 4681-4703
Abstract
We establish a probabilistic representation for a wide class of linear deterministic p.d.e.’s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution.Keywords
This publication has 19 references indexed in Scilit:
- Second-order hyperbolic s.p.d.e.’s driven by homogeneous Gaussian noise on a hyperplaneTransactions of the American Mathematical Society, 2005
- A stochastic particle numerical method for 3D Boltzmann equations without cutoffMathematics of Computation, 2001
- Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'sElectronic Journal of Probability, 1999
- The stochastic wave equation in two spatial dimensionsThe Annals of Probability, 1998
- Parabolic Anderson problem and intermittencyMemoirs of the American Mathematical Society, 1994
- Stochastic Equations in Infinite Dimensions1992
- Introduction to Stochastic Integration1990
- Local relativistic invariant flows for quantum fieldsCommunications in Mathematical Physics, 1983
- Random evolutions: A survey of results and problemsRocky Mountain Journal of Mathematics, 1974
- A stochastic model related to the telegrapher's equationRocky Mountain Journal of Mathematics, 1974