Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2

TL;DR

This paper investigates superdiffusive behavior in two models of random diffusion in two dimensions, providing bounds on variance growth and insights into their asymptotic scaling, with implications for understanding complex stochastic processes.

Contribution

The paper establishes upper and lower bounds on the variance growth of self-repelling Brownian polymers and diffusions in Gaussian free field environments, advancing understanding of their superdiffusive nature.

Findings

Variance grows faster than linearly with time.

Bounds of order t log log t and t log t for different models.

Expected true variance order is t (log t)^{1/2} and t (log t)^{2/3} for isotropic and non-isotropic cases.

Abstract

We consider two models of random diffusion in random environment in two dimensions. The first example is the self-repelling Brownian polymer, this describes a diffusion pushed by the negative gradient of its own occupation time measure (local time). The second example is a diffusion in a fixed random environment given by the curl of massless Gaussian free field. In both cases we show that the process is superdiffusive: the variance grows faster than linearly with time. We give lower and upper bounds of the order of t log log t, respectively, t log t. We also present computations for an anisotropic version of the self-repelling Brownian polymer where we give lower and upper bounds of t (log t)^{1/2}, respectively, t log t. The bounds are given in the sense of Laplace transforms, the proofs rely on the resolvent method. The true order of the variance for these processes is expected to be t (log t)^{1/2} for the isotropic and t (log t)^{2/3} for the non-isotropic case. In the appendix we present a non-rigorous derivation of these scaling exponents.