Small Ball Probabilities for the Infinite-Dimensional Ornstein-Uhlenbeck Process in Sobolev Spaces

Sergey V. Lototsky

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arXiv · 1603.08279

Small Ball Probabilities for the Infinite-Dimensional Ornstein-Uhlenbeck Process in Sobolev Spaces

Sergey V. Lototsky

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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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Sergey V. Lototsky. 2016-03-28. Small Ball Probabilities for the Infinite-Dimensional Ornstein-Uhlenbeck Process in Sobolev Spaces. https://arxiv.org/abs/1603.08279

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