The Continuum Directed Random Polymer

TL;DR

This paper constructs a continuum model of directed polymers interacting with space-time white noise, revealing that the resulting paths share Brownian properties but are singular with respect to standard Wiener measure.

Contribution

It introduces a continuum directed random polymer model driven by space-time white noise, connecting stochastic heat equations with path measures.

Findings

Paths have Brownian-like Holder continuity and quadratic variation.

The path measure is singular with respect to Wiener measure.

Model applies for all positive inverse temperature beta.

Abstract

Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter beta, and for a given beta and realization of the noise the path evolves in a Markovian way. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all beta > 0 and for almost all realizations of the white noise the path measure has the same Holder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).