Superdiffusivity for a Brownian polymer in a continuous Gaussian environment

TL;DR

This paper studies the long-term behavior of a one-dimensional Brownian polymer in a Gaussian random environment, demonstrating superdiffusive behavior with a wandering exponent exceeding 3/5.

Contribution

It provides a lower bound on the wandering exponent of the polymer, establishing superdiffusivity based on the environment's covariance structure.

Findings

Polymer exhibits superdiffusive behavior.

Wandering exponent exceeds any /5.

Results depend on the environment's covariance.

Abstract

This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field $W$ on ${\mathbb{R}}_+\times{\mathbb{R}}$ which is white noise in time and function-valued in space. According to the behavior of the spatial covariance of $W$, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any $\alpha<3/5$.