Standardizing densities on Gaussian spaces

In the present note we investigate the problem of standardizing random variables taking values on infinite dimensional Gaussian spaces. In particular, we focus on the transformations induced on densities by the selected standardization procedure. We discover that, under certain conditions, the Wick exponentials are the key ingredients for treating this kind of problems.Comment: 12 page

Nash estimates and upper bounds for non-homogeneous Kolmogorov equations

We prove a Gaussian upper bound for the fundamental solutions of a class of ultra-parabolic equations in divergence form. The bound is independent on the smoothness of the coefficients and generalizes some classical results by Nash, Aronson and Davies. The class considered has relevant applications in the theory of stochastic processes, in physics and in mathematical finance.Comment: 21 page

A note on a local limit theorem for Wiener space valued random variables

We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired L1-convergence of the density of \frac{X_1+...+X_n}{\sqrt{n}}. We close the paper comparing our result with certain Berry-Esseen bounds for multidimensional central limit theorems.Comment: 12 pages. To appear in Bernoull