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

We consider a random variable XX satisfying almost-sure conditions involving G:=〈DX,−DL−1X〉G:=〈DX,−DL−1X〉 where DXDX is XX’s Malliavin derivative and L−1L−1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. A lower- (resp. upper-) bound condition on GG 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 1x>z, and its relation to GG. 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 GG, recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for GG, 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 χ=1/2χ=1/2. We also show this exponent remains 1/21/2 after a non-linear transformation of the polymer’s Hamiltonian.