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Almost sure exponential behaviour for a parabolic SPDE on a manifold

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  • Tindel, Samy
  • Viens, Frederi

Abstract

We derive an upper bound on the large-time exponential behavior of the solution to a stochastic partial differential equation on a compact manifold with multiplicative noise potential. The potential is a random field that is white-noise in time, and Hölder-continuous in space. The stochastic PDE is interpreted in its evolution (semigroup) sense. A Feynman-Kac formula is derived for the solution, which is an expectation of an exponential functional of Brownian paths on the manifold. The main analytic technique is to discretize the Brownian paths, replacing them by piecewise-constant paths. The error committed by this replacement is controlled using Gaussian regularity estimates; these are also invoked to calculate the exponential rate of increase for the discretized Feynman-Kac formula. The error is proved to be negligible if the diffusion coefficient in the stochastic PDE is small enough. The main result extends a bound of Carmona and Viens (Stochast. Stochast. Rep. 62 (3-4) (1998) 251) beyond flat space to the case of a manifold.

Suggested Citation

  • Tindel, Samy & Viens, Frederi, 0. "Almost sure exponential behaviour for a parabolic SPDE on a manifold," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 53-74, July.
    • Handle: RePEc:eee:spapps:v:100:y::i:1-2:p:53-74
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    References listed on IDEAS

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    1. Bergé, Benjamin & D. Chueshov, Igor & Vuillermot, Pierre-A., 2001. "On the behavior of solutions to certain parabolic SPDE's driven by wiener processes," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 237-263, April.
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    Cited by:

    1. Song, Jian, 2012. "Asymptotic behavior of the solution of heat equation driven by fractional white noise," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 614-620.

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