Small Ball Probabilities for the Infinite-Dimensional Ornstein-Uhlenbeck   Process in Sobolev Spaces
  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. 
Sergey V. Lototsky
math.PR