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. |