Application of Stein's lemma and Malliavin calculus to the densities and fluctuation exponents of stochastic heat equations
Seminar Room 1, Newton Institute
When a scalar random variable X is differentiable in the sense of Malliavin with respect to an isonormal Gaussian process, we consider the random variable G := where D is the Malliavin derivative
operator, and M is the pseudo-inverse of the so-called Ornstein-Uhlenbeck semigroup generator. Like D, this G is a random way of measuring the dispersion of X; for instance, Var[X]=E[G]. Moreover, G is constant if and
only if X is Gaussian, and its use to characterize the distribution of X may be somewhat easier than D's. In this talk we will examine how the comparison of G to a constant can be used to derive, via the Malliavin calculus and/or Stein's lemma, Gaussian upper and lower bounds on the density of X. We will present applications of these results to the densities of the solutions of additive and multiplicative stochastic heat equations. In the multiplicative case, examples are identified which address a conjecture on polymer fluctuation exponents in random environments.