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

Le Chen, Cheng Ouyang, Samy Tindel, 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.

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

Abstract

We introduce and analyze a broad class of continuous directed polymers in 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 (equivalently, the noise is non-trace-class), and it is equivalent otherwise. Finally, in dimension , we prove diffusive behavior at large times in the high-temperature regime. This extends the Alberts--Khanin--Quastel framework from the white-noise setting to higher-dimensional Gaussian environments with general spatial covariance.
Paper Structure (19 sections, 34 theorems, 399 equations)

This paper contains 19 sections, 34 theorems, 399 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Discrete polymer models
  4. Continuous polymer models
  5. The stochastic heat equation
  6. The polymer measure
  7. Main results and strategy
  8. Outline of the paper
  9. Preliminary results
  10. Wiener space related to W
  11. Some properties of the stochastic heat equation
  12. Local properties of the polymer
  13. Pathwise properties
  14. Singularity of the polymer measure
  15. Diffusive behavior in high temperature
  16. ...and 4 more sections

Key Result

Theorem 1.4

Consider equation E:SHE starting from an initial condition $\mu \ge 0$, which is a nonnegative Borel measure on $\mathbb{R}^d$ such that Recall that the covariance of the noise is given by E:Cor, and is specified by a correlation measure $f$ satisfying E:Dalang. Then for any $\beta>0$, there exists a unique random field solution to equation E:SHE, interpreted as in E:mild.

Theorems & Definitions (88)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 2.2
  • ...and 78 more