The Multifractal Nature of Boltzmann Processes

TL;DR

This paper investigates the multifractal properties of the stochastic velocity process in the Boltzmann equation, revealing its complex singularity spectrum and extending analysis to the position process for hard potentials.

Contribution

It provides a novel analysis of the pathwise multifractal nature of Boltzmann processes, including explicit spectrum calculations for velocity and position.

Findings

Velocity process is almost surely multifractal.

Computed the spectrum of singularities for velocity.

Derived the multifractal spectrum for the position process in hard potentials.

Abstract

We consider the spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We study the pathwise properties of the stochastic process $(V_t)_{t\geq 0}$, which describes the time evolution of the velocity of a typical particle. We show that this process is almost surely multifractal and we compute its spectrum of singularities. For hard potentials, we also compute the multifractal spectrum of the position process $(X_t)_{t\ge0}$.