A limit theorem to a time-fractional diffusion

TL;DR

This paper establishes a limit theorem showing that the integrated force on a test particle in a periodic potential converges to a time-changed Brownian motion, revealing a fractional diffusion behavior in the limit.

Contribution

It introduces a novel limit theorem for a Markov process with Boltzmann dynamics, linking the force integral to a time-changed Brownian motion in the fractional diffusion regime.

Findings

The force integral converges to a Brownian motion time-changed by an Ornstein-Uhlenbeck local time.

The result characterizes fractional diffusion behavior in a Boltzmann-type Markov process.

Provides a mathematical foundation for anomalous diffusion in particle systems.

Abstract

We prove a limit theorem for an integral functional of a Markov process. The Markovian dynamics is characterized by a linear Boltzmann equation modeling a one-dimensional test particle of mass $\lambda^{-1}\gg 1$ in an external periodic potential and undergoing collisions with a background gas of particles with mass one. The object of our limit theorem is the time integral of the force exerted on the test particle by the potential, and we consider this quantity in the limit that $\lambda$ tends to zero for time intervals on the scale $\lambda^{-1}$. Under appropriate rescaling, the total drift in momentum generated by the potential converges to a Brownian motion time-changed by the local time at zero of an Ornstein-Uhlenbeck process.