A class of d-dimensional directed polymers in a Gaussian environment

TL;DR

This paper studies a broad class of directed polymers in Gaussian environments across multiple dimensions, establishing their structural properties, path behavior, measure singularity conditions, and diffusive behavior in high dimensions.

Contribution

It extends the framework of directed polymers from 1+1 white-noise to higher-dimensional Gaussian environments with general spatial covariance, analyzing their properties and behavior.

Findings

Partition function has positivity, stationarity, and scaling properties.

Polymer paths exhibit Hölder continuity and quadratic variation identification.

In high dimensions, polymers show diffusive behavior at large times in high-temperature regimes.

Abstract

We introduce and analyze a broad class of continuous directed polymers in $\mathbb{R}^d$ driven by Gaussian environments that are white in time and spatially correlated, under Dalang's condition. Using an It\^o-renormalized stochastic-heat-equation representation, we establish structural properties of the partition function, including positivity, stationarity, scaling, homogeneity, and a Chapman--Kolmogorov relation. On finite time intervals, we prove Brownian-type pathwise behavior, namely H\"older continuity and identification of the quadratic variation. We then obtain a sharp measure-theoretic dichotomy: the quenched polymer measure is singular with respect to Wiener measure if and only if $\widehat f(\mathbb{R}^d)=\infty$ (equivalently, the noise is non-trace-class), and it is equivalent otherwise. Finally, in dimension $d\ge 3$, we prove diffusive behavior at large times in the high-temperature regime. This extends the Alberts--Khanin--Quastel framework from the $1+1$ white-noise setting to higher-dimensional Gaussian environments with general spatial covariance.