Interpolation process between standard diffusion and fractional diffusion

TL;DR

This paper studies a stochastic lattice model with two conserved quantities, showing how energy fluctuations transition from fractional to normal diffusion, and identifies a critical noise intensity where fluctuations follow an Ornstein-Uhlenbeck process driven by a Lévy process interpolating between Brownian motion and a 3/2-stable Lévy process.

Contribution

It characterizes the energy fluctuation process at a critical noise intensity, revealing an interpolation between Brownian and Lévy stable processes, extending previous open problems.

Findings

Energy fluctuations are described by an Ornstein-Uhlenbeck process driven by a Lévy process.

At a critical noise intensity, the process interpolates between Brownian motion and a 3/2-stable Lévy process.

Normal diffusion is restored at high noise intensity, while fractional diffusion persists at low intensity.

Abstract

We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume [5, 3]. We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper we investigate the nature of the energy fluctuations for a critical value of the intensity. We show that the latter are described by an Ornstein-Uhlenbeck process driven by a L\'evy process which interpolates between Brownian motion and the maximally asymmetric 3/2-stable L\'evy process. This result extends and solves a problem left open in [4].