|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 ), and on the density of X
via a new density formula (see ). A connection to Stein's lemma,
particularly in , can be extended to comparisons with other
distributions than the normal, including the entire so-called Pearson
class, as seen in . A multidimensional extension of the density
formula is obtained in , 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  and .
 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.
 Eden, R.; Viens, F. General upper and lower tail estimates using
Malliavin calculus and Stein's equations. Preprint, 2010.
 Nourdin, I; Viens, F. Density estimates and concentration inequalities
with Malliavin calculus. Electronic Journal of Probability, 14 (2009),
 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.
 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.|