Regularity and Stability for the Semigroup of Jump Diffusions with State-Dependent Intensity

TL;DR

This paper studies the regularity and stability of semigroups generated by jump diffusion processes with state-dependent jump intensities, providing conditions for regularity preservation and explicit stability estimates, with applications to physical models.

Contribution

It offers new conditions ensuring regularity preservation and explicit stability bounds for semigroups of jump diffusions with state-dependent intensities, including applications to the Boltzmann equation.

Findings

Semigroup preserves regularity under certain conditions.

Explicit operator norm estimates are derived.

Stability estimates quantify differences between semigroups.

Abstract

We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc). First, we give sufficient conditions guaranteeing that the semigroup associated with such an equation preserves regularity by mapping the space of the of k-times differentiable bounded functions into itself. Furthermore, we give an explicit estimate of the operator norm. This is the key-ingredient in a quantitative Trotter-Kato-type stability result: it allows us to give an explicit estimate of the distance between two semigroups associated with different sets of coefficients in terms of the difference between the corresponding infinitesimal operators. As an application, we present a method allowing to replace " small jumps " by a Brownian motion or by a drift component. The example of the 2D Boltzmann equation is also treated in all detail.