Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent

Abstract

We consider a random variable X satisfying almost-sure conditions involving G:=<DX,-DL^{-1}X> where DX is X's Malliavin derivative and L^{-1} is the inverse Ornstein-Uhlenbeck operator. A lower- (resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P[X>z]. Bounds of other natures are also given. A key ingredient is the use of Stein's lemma, including the explicit form of the solution of Stein's equation relative to the function 1_{x>z}, and its relation to G. Another set of comparable results is established, without the use of Stein's lemma, using instead a formula for the density of a random variable based on G, recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for G, we show that the Brownian polymer in a Gaussian environment which is white-noise in time and positively correlated in space has deviations of Gaussian type and a fluctuation exponent \chi=1/2. We also show this exponent remains 1/2 after a non-linear transformation of the polymer's Hamiltonian.