Stochastic differential equations for infinite particle systems of jump type with long range interactions

TL;DR

This paper studies infinite-dimensional stochastic differential equations modeling systems of infinitely many particles undergoing Lévy processes with long-range interactions, establishing conditions for their strong solutions' existence and uniqueness.

Contribution

It introduces a framework for analyzing ISDEs with Lévy processes and long-range potentials, proving existence and uniqueness of strong solutions.

Findings

Existence of strong solutions for the ISDEs.

Uniqueness of solutions under specified conditions.

Applicability to systems with Ruelle's class or logarithmic potentials.

Abstract

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a L\'evy process, and the interaction between particles is determined by the long-range interaction potential. The potential is of Ruelle's class or logarithmic. We discuss the existence and uniqueness of strong solutions of the ISDEs.