Annals of Probability

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

Yaozhong Hu, David Nualart, and Jian Song

Full-text: Open access


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

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

First available in Project Euclid: 3 December 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

Export citation


  • [1] Dawson, D. A. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141–180.
  • [2] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Annals of Mathematics Studies 109. Princeton Univ. Press, Princeton, NJ.
  • [3] Hinz, H. (2009). Burgers system with a fractional Brownian random force. Preprint, Technische Univ. Berlin.
  • [4] Hu, Y., Lu, F. and Nualart, D. (2010). Feynman–Kac formula for the heat equation driven by fractional noise with Hurst parameter H<1∕2. Preprint, Univ. Kansas.
  • [5] Hu, Y. and Nualart, D. (2009). Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 285–328.
  • [6] Hu, Y.-Z. and Yan, J.-A. (2009). Wick calculus for nonlinear Gaussian functionals. Acta Math. Appl. Sin. Engl. Ser. 25 399–414.
  • [7] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [8] Le Gall, J.-F. (1994). Exponential moments for the renormalized self-intersection local time of planar Brownian motion. In Séminaire de Probabilités, XXVIII. Lecture Notes in Math. 1583 172–180. Springer, Berlin.
  • [9] Mocioalca, O. and Viens, F. (2005). Skorohod integration and stochastic calculus beyond the fractional Brownian scale. J. Funct. Anal. 222 385–434.
  • [10] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [11] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421.
  • [12] Viens, F. G. and Zhang, T. (2008). Almost sure exponential behavior of a directed polymer in a fractional Brownian environment. J. Funct. Anal. 255 2810–2860.
  • [13] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint–Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.