Bounds for the state-modulated resolvent of a linear Boltzmann generator

TL;DR

This paper derives bounds for a state-modulated resolvent of a Markovian generator in a linear Boltzmann model, focusing on regimes with large mass ratios, to understand particle phase space dynamics.

Contribution

It provides new bounds for the state-modulated resolvent of a linear Boltzmann generator in the large mass ratio regime, linking to phase space residence times.

Findings

Bounds are established for the resolvent in the large mass ratio regime.

Results relate the resolvent bounds to the time spent in phase space regions.

Insights into particle behavior in the linear Boltzmann framework.

Abstract

We study a generalized resolvent for the generator of a Markovian semigroup. The Markovian generator appears in a linear Boltzmann equation modeling a one-dimensional test particle in a periodic potential and colliding elastically with particles from an ideal background gas. We obtain bounds for the state-modulated resolvent which are relevant in the regime where the mass ratio between the test particle and a particle from the gas is large. These bounds relate to the typical amount of time that the particle spends in different regions of phase space before arriving to a region around the origin.