The Annals of Probability

Feynman–Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2

Yaozhong Hu, Fei Lu, and David Nualart

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 40, Number 3 (2012), 1041-1068.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1336136058

Digital Object Identifier
doi:10.1214/11-AOP649

Mathematical Reviews number (MathSciNet)
MR2962086

Zentralblatt MATH identifier
1253.60074

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G22: Fractional processes, including fractional Brownian motion 60H05: Stochastic integrals 60H30: Applications of stochastic analysis (to PDE, etc.) 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

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 &lt; 1/2. Ann. Probab. 40 (2012), no. 3, 1041--1068. doi:10.1214/11-AOP649. https://projecteuclid.org/euclid.aop/1336136058


Export citation

References

  • [1] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Statist. Phys. 78 1377–1401.
  • [2] Hu, Y., Jolis, M. and Tindel, S. (2011). On Stratonovich and Skorohod stochastic calculus for Gaussian processes. Preprint. Available at http://arxiv.org/abs/1101.3441.
  • [3] Hu, Y. and Nualart, D. (2009). Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 285–328.
  • [4] Hu, Y., Nualart, D. and Song, J. (2011). Feynman–Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39 291–326.
  • [5] Kruk, I. and Russo, F. (2010). Malliavin–Skorohod calculus and Paley–Wiener integral for covariance singular processes. Preprint. Available at http://arxiv.org/abs/1011.6478.
  • [6] Mocioalca, O. and Viens, F. (2005). Skorohod integration and stochastic calculus beyond the fractional Brownian scale. J. Funct. Anal. 222 385–434.
  • [7] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [8] Nualart, D. and Schoutens, W. (2001). Backward stochastic differential equations and Feynman–Kac formula for Lévy processes, with applications in finance. Bernoulli 7 761–776.
  • [9] Ocone, D. and Pardoux, É. (1993). A stochastic Feynman–Kac formula for anticipating SPDEs, and application to nonlinear smoothing. Stochastics Stochastics Rep. 45 79–126.
  • [10] Ouerdiane, H. and Silva, J. L. (2002). Generalized Feynman–Kac formula with stochastic potential. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 243–255.
  • [11] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon.
  • [12] Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 333–374.