The Annals of Probability

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

Yaozhong Hu, David Nualart, and Jian Song

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Abstract

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 Hölder 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.

Article information

Source
Ann. Probab., Volume 39, Number 1 (2011), 291-326.

Dates
First available in Project Euclid: 3 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1291388303

Digital Object Identifier
doi:10.1214/10-AOP547

Mathematical Reviews number (MathSciNet)
MR2778803

Zentralblatt MATH identifier
1210.60056

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H15: Stochastic partial differential equations [See also 35R60] 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: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
Fractional noise stochastic heat equations Feynman–Kac formula exponential integrability absolute continuity Hölder continuity chaos expansion

Citation

Hu, Yaozhong; Nualart, David; Song, Jian. Feynman–Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39 (2011), no. 1, 291--326. doi:10.1214/10-AOP547. https://projecteuclid.org/euclid.aop/1291388303


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