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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 integrabiliy. 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 Skorohod-type equation. They also get a Feynman-Kac formula for this solution.

MSC:
60H07Stochastic calculus of variations and the Malliavin calculus
60H15Stochastic partial differential equations
60G17Sample path properties
60G22Fractional processes, including fractional Brownian motion
60G30Continuity and singularity of induced measures (stochastic processes)
35K20Second order parabolic equations, initial boundary value problems
35R60PDEs with randomness, stochastic PDE