41. chen.ouyang.ea:26:class
^^^^^^^^^^^^^^^^^^^^^^^^^^

.. centered:: A class of d-dimensional directed polymers in a Gaussian environment
.. centered:: Le Chen, Cheng Ouyang, Samy Tindel, and Panqiu Xia

**Abstract:** We introduce and analyze a broad class of continuous directed
polymers in :math:`\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
:math:`\widehat{f}(\mathbb{R}^d)=\infty` (equivalently, the noise is
non-trace-class), and it is equivalent otherwise. Finally, in dimension
:math:`d\ge 3`, we prove diffusive behavior at large times in the
high-temperature regime. This extends the Alberts--Khanin--Quastel framework
from the :math:`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.

`Preprint <https://arxiv.org/abs/2603.06574>`_

:cite:`chen.ouyang.ea:26:class` 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*

.. code-block:: bibtex

   @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}
   }


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