Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent
| dc.creator | Viens, Frederi G. | |
| dc.date | 2009-01-04 | |
| dc.date.accessioned | 2026-07-07T12:24:33Z | |
| dc.date.available | 2026-07-07T12:24:33Z | |
| dc.description | 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 χ=1/2. We also show this exponent remains 1/2 after a non-linear transformation of the polymer's Hamiltonian. | |
| dc.description | 24 pages | |
| dc.identifier | https://arxiv.org/abs/0901.0383 | |
| dc.identifier | http://arxiv.org/abs/0901.0383 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/214357 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60H07; 60G15; 60K37; 82D60 | |
| dc.title | Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent | |
| dc.type | text |