Consider a centered random variable $X$ on a Wiener space, satisfying almost-sure conditions involving $G:=$ where $DX$ is $X$'s Malliavin derivative and $M$ is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. It turns out that $X$ is standard normal if and only if $G=1$ almost surely. Nourdin and Peccati developed a machinery which shows how to estimate the distance between the law of $X$ and the standard normal law, when $G$ is close to $1$ in an appropriate sense. In this talk we present one-sided comparisons of $G$ to the constant $1$ resulting in Gaussian-type lower and upper bounds on the tail $P[X>z],$ with applications to functionals of Gaussian fields such as suprema. Extensions of these types of results to comparisons beyond the Gaussian scale are also established, with particular application to the Pearson class, thanks to a study of the solutions of relevant Stein equations. We also show how a new formula for the density of $X$ based on $G$ implies a convenient framework for establishing Gaussian upper and lower bounds on this density, with applications to stochastic PDEs. A multidimensional extension of the density formula is available, although Gaussian comparisons in this case, particularly lower bounds, still elude us. This talk covers works by H. Airault, R.Eden, P. Malliavin, I. Nourdin, G. Peccati, and the presenter.
Speaker: Frederi Viens, Purdue University