Regularization and Asymptotic Behaviour of Ornstein-Uhlenbeck Evolution Operators in Infinite Dimension

We are concerned with the properties of Ornstein-Uhlenbeck evolution operators acting on functions defined in a Hilbert space and p-integrable with respect to a suitable Gaussian measure.

These operators provide solutions to the infinite dimensional, non-autonomous backward Kolmogorov equation.

The first part of the paper focuses on some general regularization properties, while the second part carries on a deep analysis of the asymptotic behaviour in the periodic case.

In particular, we identify the optimal convergence rate of the Ornstein-Uhlenbeck operator with respect to time and we give an optimality criterion depending only on the drift term of the Kolmogorov equation.