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

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 Itô-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 Hölder 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 $\hat{f}({\mathbb{R}}^d)=\infty$ (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.

68 pages