41. chen.ouyang.ea:26:class#
A class of d-dimensional directed polymers in a Gaussian environment
Le Chen, Cheng Ouyang, Samy Tindel, and Panqiu Xia
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 \(\widehat{f}(\mathbb{R}^d)=\infty\) (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.
MSC 2020 subject classifications: Primary 60H15; Secondary 60K37, 82D60, 60G15.
Keywords: directed polymers, Gaussian environment, stochastic heat equation, Dalang’s condition, partition function, quenched measure, Wiener measure, diffusive behavior, high-temperature regime.
[COTX26] Le Chen, Cheng Ouyang, Samy Tindel & Panqiu Xia (2026) ‘A class of d-dimensional directed polymers in a Gaussian environment’, preprint arXiv:2603.06574, 68 pages
@article{chen.ouyang.ea:26:class,
title = {A class of d-dimensional directed polymers in a Gaussian environment},
author = {Le Chen and Cheng Ouyang and Samy Tindel and Panqiu Xia},
year = {2026},
month = {March},
journal = {Preprint arXiv:2603.06574},
url = {http://arXiv.org/abs/2603.06574}
}