The intermediate disorder regime for Brownian directed polymers in Poisson environment

Abstract

We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime [2], which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener-Itô chaos expansion [38], which, applied to the partition function, is the basic ingredient of the proof.AMS 2010 subject classifications. 60F05, 82D60.

Zero temperature limit for the Brownian directed polymer among Poissonian disasters

“…Remark 1.1. In the earlier works [9,10,8,6,11] on Brownian directed polymers, there is no negative sign in front of β in the definitions (1.1) and (1.2) and general β ∈ R is considered. We use the above formulation since we focus on the repulsive case in this article.…”