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Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2
Hu, Yaozhong Lu, Fei Nulart, David
Hu, Yaozhong
Lu, Fei
Nulart, David
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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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This is the published version, also available here: http://dx.doi.org/10.1214/11-AOP649.
Date
2012-09-01
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Institute of Mathematical Statistics
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Keywords
Feynman-Kac integral, Feynman-Kac formula, stochastic partial differential equations, fractional Brownian field, nonlinear stochastic integral, fractional calculus
Citation
Hu, Yaozhong., Lu, Fei., Nualart, David. "Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2." Ann. Probab. Volume 40, Number 3 (2012), 1041-1068. http://dx.doi.org/10.1214/11-AOP649.