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

TL;DR

This paper derives precise asymptotics for small ball probabilities of the infinite-dimensional Ornstein-Uhlenbeck process in Sobolev spaces, revealing new effects compared to finite-dimensional cases.

Contribution

It provides exact logarithmic asymptotics for small ball probabilities of solutions to stochastic parabolic equations in Sobolev spaces, extending understanding beyond finite-dimensional scenarios.

Findings

Exact logarithmic asymptotics derived

New effects identified in certain Sobolev exponent ranges

Results applicable to Gaussian measures from stochastic PDE solutions

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.