Meeting Details

Title: Malliavin calculus, density estimates, and Stein's lemma. Seminar on Probability and its Application Frederi G. Viens, Purdue University Consider a centered random variable X satisfying almost-sure conditions involving G : =(DX; -DMX) 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 Gaussian if and only if G=1 almost surely. The comparison of G with the constant 1 allow one to show Gaussian-type lower and upper bounds on the tail P[X > z] (see [5]), and on the density of X via a new density formula (see [3]). A connection to Stein's lemma, particularly in [5], can be extended to comparisons with other distributions than the normal, including the entire so-called Pearson class, as seen in [2]. A multidimensional extension of the density formula is obtained in [1], although Gaussian comparisons in this case, particularly lower bounds, still elude us. Time permitting, we will mention some examples where the estimation of G is relatively straightforward, via a Malliavin-calculus device based on the Mehler formula, resulting in some striking applications to stochastic PDEs, as in [4] and [5]. Literature: [1] Airault, H.; Malliavin, P.; Viens, F. Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis. Journal of Functional Analysis, 258 no. 5 (2009), 1763-1783. [2] Eden, R.; Viens, F. General upper and lower tail estimates using Malliavin calculus and Stein's equations. Preprint, 2010. [3] Nourdin, I; Viens, F. Density estimates and concentration inequalities with Malliavin calculus. Electronic Journal of Probability, 14 (2009), 2287-2309. [4] Nualart, D.; Quer-Sardayons, Ll. Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations. To appear in Stochastic Processes and their Applications, 2010. [5] Viens, F. Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent. Stochastic Processes and their Applications 119 (2009), 3671-3698.

Room Reservation Information

Room Number: MB106 04 / 15 / 2011 02:20pm - 03:20pm