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May 2012 Feynman–Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2
Yaozhong Hu, Fei Lu, David Nualart
Ann. Probab. 40(3): 1041-1068 (May 2012). DOI: 10.1214/11-AOP649

Abstract

In this paper, a Feynman–Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H < 1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman–Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malliavin calculus is used to show that the Feynman–Kac integral is the weak solution to the stochastic partial differential equation.

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Yaozhong Hu. Fei Lu. David Nualart. "Feynman–Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2." Ann. Probab. 40 (3) 1041 - 1068, May 2012. https://doi.org/10.1214/11-AOP649

Information

Published: May 2012
First available in Project Euclid: 4 May 2012

zbMATH: 1253.60074
MathSciNet: MR2962086
Digital Object Identifier: 10.1214/11-AOP649

Subjects:
Primary: 35R60 , 60G22 , 60H05 , 60H15 , 60H30

Keywords: Feynman–Kac formula , Feynman–Kac integral , fractional Brownian field , Fractional calculus , nonlinear stochastic integral , Stochastic partial differential equations

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 3 • May 2012
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