Feynman-Kac formula for heat equation driven by fractional white noise

Details Feynman-Kac formula for heat equation driven by fractional white noise Feynman-Kac formula for heat equation driven by fractional white noise

2009-06-17

arxiv.org/abs/0906.3076v2

Annals of Probability 2011, Vol. 39, No. 1, 291-326

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP547 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP547

We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman-Kac formula is a weak solution of the stochastic heat equation. From the Feynman-Kac formula, we establish the smoothness of the density of the solution and the H\"{o}lder regularity in the space and time variables. We also derive a Feynman-Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.

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