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Small ball probabilities for the infinite-dimensional Ornstein–Uhlenbeck process in Sobolev spaces

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Abstract

While small ball, or lower tail, asymptotic for Gaussian measures generated by solutions of stochastic ordinary differential equations is relatively well understood, a lot less is known in the case of stochastic partial differential equations. The paper presents exact logarithmic asymptotics of the small ball probabilities in a scale of Sobolev spaces when the Gaussian measure is generated by the solution of a diagonalizable stochastic parabolic equation. Compared to the finite-dimensional case, new effects appear in a certain range of the Sobolev exponents.

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  1. Department of Mathematics, USC, Los Angeles, CA, 90089, USA

    S. V. Lototsky

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  1. S. V. Lototsky
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Lototsky, S.V. Small ball probabilities for the infinite-dimensional Ornstein–Uhlenbeck process in Sobolev spaces. Stoch PDE: Anal Comp 5, 192–219 (2017). https://doi.org/10.1007/s40072-016-0085-y

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  • DOI: https://doi.org/10.1007/s40072-016-0085-y

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