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Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs

Wei Sun, Ethan Wong

TL;DR

The paper addresses the existence of periodic and stationary solutions for distribution-dependent SDEs (DDSDEs) where coefficients depend on the law of the solution. It develops a hybrid approach combining weak convergence (Kurtz–Markov, Prokhorov) and Lyapunov drift criteria to obtain existence criteria under three Lyapunov-type conditions (H3a)-(H3c) and a Schauder fixed point argument. A central result is that a $\theta$-periodic measure in ${\mathcal{P}}_{\vartheta}({\mathbb{R}}^d)$ yields a $\theta$-periodic solution to the DDSDE, with time-homogeneous cases producing invariant measures; non-uniqueness of such measures is not excluded. The paper also provides concrete, time-periodic interaction examples that illustrate the applicability of the criteria to nonlinear DDSDEs, including convolution-type and Landau-type models.

Abstract

We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and Lyapunov functions can be combined to give efficient criteria for the existence of periodic and stationary solutions. Concrete examples are presented to illustrate the novel criteria.

Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs

TL;DR

The paper addresses the existence of periodic and stationary solutions for distribution-dependent SDEs (DDSDEs) where coefficients depend on the law of the solution. It develops a hybrid approach combining weak convergence (Kurtz–Markov, Prokhorov) and Lyapunov drift criteria to obtain existence criteria under three Lyapunov-type conditions (H3a)-(H3c) and a Schauder fixed point argument. A central result is that a -periodic measure in yields a -periodic solution to the DDSDE, with time-homogeneous cases producing invariant measures; non-uniqueness of such measures is not excluded. The paper also provides concrete, time-periodic interaction examples that illustrate the applicability of the criteria to nonlinear DDSDEs, including convolution-type and Landau-type models.

Abstract

We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and Lyapunov functions can be combined to give efficient criteria for the existence of periodic and stationary solutions. Concrete examples are presented to illustrate the novel criteria.
Paper Structure (3 sections, 5 theorems, 79 equations)

This paper contains 3 sections, 5 theorems, 79 equations.

Table of Contents

  1. Introduction and Main Results
  2. Proof of Theorem \ref{['thm1']}
  3. Examples

Key Result

Theorem 1.3

Let $\vartheta \in [1,\infty)$. Suppose that conditions (H0)--(H2) hold. Then, there exists a $\theta$-periodic solution to the DDSDE (DDSDE) if one of the following three conditions holds.

Theorems & Definitions (18)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Corollary 1.5
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 3.1
  • ...and 8 more