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Random periodic solutions for stochastic differential equations with non-uniform dissipativity

Jianhai Bao, Yue Wu

TL;DR

The paper addresses the existence and uniqueness of random periodic solutions for stochastic systems with non-uniform dissipativity. It develops two complementary approaches: reflection coupling to obtain random periodic solutions in distribution for SDEs without memory under dissipativity at long distance, and synchronous coupling to obtain pathwise random periodic solutions for functional SDEs with finite or infinite lag under dissipativity on average. The results are stated under integral dissipativity conditions involving time-periodic coefficients and BDG constants, and include both SDEs with additive noise and functional SDEs with memory. Together, they significantly broaden the scope of random periodic solution theory beyond uniformly dissipative settings and provide robust criteria for existence, uniqueness, and pullback convergence to periodic states with potential implications for long-term behavior in stochastic systems with memory.

Abstract

This paper is concerned with the existence and uniqueness of random periodic solutions for stochastic differential equations (SDEs), where the drift terms involved need not to be uniformly dissipative. On the one hand, via the reflection coupling approach, we investigate the existence of random periodic solutions in the sense of distribution for SDEs without memory, where the drifts are merely dissipative at long distance. On the other hand, via the synchronous coupling strategy, we establish respectively the existence of pathwise random periodic solutions for functional SDEs with a finite time lag and an infinite time lag, in which the drifts are only dissipative on average rather than uniformly dissipative with respect to the time parameters.

Random periodic solutions for stochastic differential equations with non-uniform dissipativity

TL;DR

The paper addresses the existence and uniqueness of random periodic solutions for stochastic systems with non-uniform dissipativity. It develops two complementary approaches: reflection coupling to obtain random periodic solutions in distribution for SDEs without memory under dissipativity at long distance, and synchronous coupling to obtain pathwise random periodic solutions for functional SDEs with finite or infinite lag under dissipativity on average. The results are stated under integral dissipativity conditions involving time-periodic coefficients and BDG constants, and include both SDEs with additive noise and functional SDEs with memory. Together, they significantly broaden the scope of random periodic solution theory beyond uniformly dissipative settings and provide robust criteria for existence, uniqueness, and pullback convergence to periodic states with potential implications for long-term behavior in stochastic systems with memory.

Abstract

This paper is concerned with the existence and uniqueness of random periodic solutions for stochastic differential equations (SDEs), where the drift terms involved need not to be uniformly dissipative. On the one hand, via the reflection coupling approach, we investigate the existence of random periodic solutions in the sense of distribution for SDEs without memory, where the drifts are merely dissipative at long distance. On the other hand, via the synchronous coupling strategy, we establish respectively the existence of pathwise random periodic solutions for functional SDEs with a finite time lag and an infinite time lag, in which the drifts are only dissipative on average rather than uniformly dissipative with respect to the time parameters.
Paper Structure (8 sections, 13 theorems, 136 equations)

This paper contains 8 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction and main results
  2. Random periodic solution for SDEs: dissipativity at long distance
  3. Random periodic solutions for functional SDEs: dissipativity on average
  4. Functional SDEs with a finite time lag: dissipativity on average
  5. Functional SDEs with an infinite time lag: dissipativity on average
  6. Proof of Theorem \ref{['main2']}
  7. Proof of Theorem \ref{['thm:main1']}
  8. Appendix

Key Result

Theorem 1.1

Under Assumption $({\bf A})$, the stochastic semi-flow $\phi$, defined by 0j, has a unique random $\tau$-periodic solution in the sense of distribution, i.e., there exists a stochastic process $(X^*(t))_{t\in\mathbb R}\in L^1(\Omega\rightarrow\mathbb R^d,\mathscr F,\mathbb P)$ such that for all $(t, and

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • proof
  • ...and 21 more