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Zbl 1210.60056
Hu, Yaozhong; Nualart, David; Song, Jian
Feynman-Kac formula for heat equation driven by fractional white noise. (English)
[J] Ann. Probab. 39, No. 1, 291-326 (2011). ISSN 0091-1798

The aim of this paper is to obtain a Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. \par 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. \par 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.
[Carles Rovira (Barcelona)]

MSC 2000:: *60H07 Stochastic calculus of variations and the Malliavin calculus
60H15 Stochastic partial differential equations
60G17 Sample path properties
60G22
60G30 Induced measures of stochastic processes
35K20 Second order parabolic equations, boundary value problems
35R60 PDE with randomness

Keywords: Fracional noise; stochastic heat equtions; Feynman-Kac formula; exponential integrability; absolute continuity; Hölder continuity; chaos expansion

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