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

Syota Esaki, Hideki Tanemura

Abstract

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a Lévy 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.

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

Abstract

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a Lévy 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.

Paper Structure

This paper contains 18 sections, 43 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statement of results
  4. Tagged particle processes and their consistency
  5. ISDE representation
  6. Existence and uniqueness of strong solutions
  7. Examples
  8. Proofs of Theorems
  9. Proofs of Theorems \ref{['T:31a']} and \ref{['T:31b']}
  10. Proof of Theorem \ref{['T:31c']}
  11. Proof of Theorem \ref{['T:32a']}
  12. Proof of Theorems \ref{['T:33a']} and \ref{['T:33b']}
  13. Sufficient conditions for assumptions
  14. Sufficient conditions for ()[NCL]
  15. Sufficient conditions for ()[NEX]
  16. ...and 3 more sections

Key Result

Lemma 2.1

(E.19) Assume that ()[A.1] and ()[A.2] hold. Let $({\mathcal{E}^{\mu}}, \mathcal{D}^{\mu})$ be the closure of $({\mathcal{E}^{\mu}}, \mathcal{D}_{\circ}^{\mu})$ on $L^2(\mathfrak{M}, \mu)$. Then, we have the following.

Theorems & Definitions (81)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Remark 1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 2
  • Theorem 3.4
  • Definition 3.5
  • ...and 71 more