Elements of the Stochastic Calculus for a Class of Boltzmann type processes and Application to Regularity of Densities

TL;DR

This paper develops a non-Gaussian stochastic calculus for certain Markov processes and applies it to study the smoothness of densities in Boltzmann-type particle systems.

Contribution

It introduces a novel stochastic calculus for piecewise deterministic Markov processes, extending Malliavin calculus to non-Gaussian settings.

Findings

Established regularity results for densities of Boltzmann-type processes

Extended stochastic calculus tools to non-Gaussian processes

Provided new methods for analyzing particle system densities

Abstract

For a class of piecewise deterministic Markov processes we introduce a stochastic calculus which is a certain non-Gaussian counterpart to the classical Malliavin calculus. As an application we investigate the regularity of densities of $N$-particle Boltzmann type processes at time $t>0$.