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Feynman-Kac formula for heat equation driven by fractional white noise. (English) Zbl 1210.60056

The aim of this paper is to obtain a Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet.
Using an approximation of the Dirac delta function, they show that the stochastic Feynman-Kac functional is a well-defined random variable with exponential integrability. Using again an approximation technique together with Malliavin calculus, they prove that the process defined by the Feynman-Kac functional is a weak solution of the stochastic heat equation. Using the explicit form of the weak solution they prove the Hölder continuity of the solution with respect time and space and they establish the smoothness of the density of the probability law of the solution.
Finally, using a wiener chaos technique, they prove that there exists a unique mild solution to the Skorokhod-type equation. They also get a Feynman-Kac formula for this solution.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G17 Sample path properties
60G22 Fractional processes, including fractional Brownian motion
60G30 Continuity and singularity of induced measures
35K20 Initial-boundary value problems for second-order parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
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References:

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