A Note on a Feynman-Kac-Type Formula

Raluca M Balan (University of Ottawa)


In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008) ,which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.

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Pages: 252-260

Publication Date: June 25, 2009

DOI: 10.1214/ECP.v14-1468


  1. R.J. Adler, D. Monrad, R.H. Scissors and R.J. Wilson. Representations, decompositions, and sample function continuity of random fields with independent increments. Stoch. Proc. Appl. 15 (1983), 3-30. Math. Review MR694534
  2. R.M. Balan and C.A. Tudor. The stochastic heat equation with fractional-colored noise: existence of the solution. Latin Amer. J. Probab. Math. Stat. 4 (2008), 57-87. Math. Review MR2413088
  3. R.M. Balan and C.A. Tudor. Stochastic heat equation with multiplicative fractional-colored noise. Preprint (2008). Math. Review number not available. arXiv:0812.1913.
  4. R.A. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Memoirs Amer. Math. Soc. 108 (1994), no. 518, viii+125 pp. Math. Review MR1185878
  5. R.A. Carmona and F. Viens. Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter. Stoch. Stoch. Rep. 62 (1998), 251-273. Math. Review MR1615092
  6. M. Cranston, T.S. Mountford and T. Shiga. Lyapunov exponent for the parabolic Anderson model with L'evy noise. Probab. Th. Rel. Fields 132 (2005), 321-355. Math. Review MR2197105
  7. R.C. Dalang. Extending martingale measure stochastic integral with application to spatially homogenous s.p.d.e.'s. Electr. J. Probab. 4 (1999), paper 6, 1-29. Math. Review MR1684157
  8. R.C. Dalang, C. Mueller and R. Tribe. A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.'s. Trans. AMS 360 (2008), 4681-4703. Math. Review MR2403701
  9. T. Deck, S. Kruse, J. Potthoff and H. Watanabe. White noise approach to s.p.d.e.'s. In: Stochastic partial differential equations and applications V, (Trento, 2002). Eds. G. Da Prato and L. Tubaro. Lecture Notes in Pure and Appl. Math., 227 (2002), 183-195. Dekker, New York. Math. Review MR1919509
  10. R. Hersch. Random evolutions: a survey of results and problems. Rocky Mountain J. Math. 4 (1974), 443-477. Math. Review MR0394877
  11. Y. Hu. Heat equations with fractional white noise potentials. Appl. Math. Optim. 43 (2001), 221-243. Math. Review MR1885698
  12. Y. Hu and D. Nualart. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Rel. Fields 143 (2009), 285-328. Math. Review MR2449130
  13. M.A. Kac. A stochastic model related to the telegraph's equation. Rocky Mountain J. Math. 4 (1974), 497-509. Math. Review MR0510166
  14. I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, Second Edition (1991). Springer, New York. Math. Review MR1121940
  15. B. Oksendal, G. Vage and H.Z. Zhao. Asymptotic properties of the solutions to stochastic KPP equations. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 1363-1381. Math. Review MR1809108
  16. M.A. Pimsky. Lectures on Random Evolution (1991). World Scientific. Math. Review MR1143780
  17. S. Tindel and F. Viens. Almost sure exponential behavior for a parabolic SPDE on a manifold. Stoch. Proc. Appl. 100 (2002), 53-74. Math. Review MR1919608

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