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 : = 〈 D X , − D L − 1 X 〉 where D X is X ’s Malliavin derivative and L − 1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. 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 χ = 1 / 2 . We also show this exponent remains 1 / 2 after a non-linear transformation of the polymer’s Hamiltonian.