REGULARITY AND STABILITY FOR THE SEMIGROUP OF JUMP DIFFUSIONS WITH STATE-DEPENDENT INTENSITY

• 成果类型: Article
• 署名作者: Bally, Vlad; Goreac, Dan; Rabiet, Victor
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Universite Gustave-Eiffel
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1382
• 发表日期: 2018
• 页码: 3028-3074
• 关键词: homogeneous boltzmann-equation; stochastic differential-equations; piecewise deterministic processes; asymptotic analysis; grazing collisions; MARKOV-PROCESSES; Levy processes; systems; approximations; networks

摘要

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.