## The Annals of Applied Probability

### Regularity and stability for the semigroup of jump diffusions with state-dependent intensity

#### 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 $k$-times differentiable bounded functions into itself. Furthermore, we give an upper estimate of the operator norm. This is the key-ingredient in a quantitative Trotter–Kato-type stability result: it allows us to give an upper 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.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 3028-3074.

Dates
Revised: December 2017
First available in Project Euclid: 28 August 2018

https://projecteuclid.org/euclid.aoap/1535443241

Digital Object Identifier
doi:10.1214/18-AAP1382

Mathematical Reviews number (MathSciNet)
MR3847980

Zentralblatt MATH identifier
06974772

#### Citation

Bally, Vlad; Goreac, Dan; Rabiet, Victor. Regularity and stability for the semigroup of jump diffusions with state-dependent intensity. Ann. Appl. Probab. 28 (2018), no. 5, 3028--3074. doi:10.1214/18-AAP1382. https://projecteuclid.org/euclid.aoap/1535443241

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