[Paper Review] Poisson stable solutions for stochastic differential equations with Lévy noise

Xin Liu, Zhenxin Liu|arXiv (Cornell University)|Feb 2, 2020

Stability and Controllability of Differential Equations30 references4 citations

TL;DR

This paper establishes the existence and global asymptotic stability of Poisson stable solutions—such as periodic, almost periodic, and Levitan almost periodic—for semilinear stochastic differential equations driven by infinite-dimensional Lévy noise with large jumps. Under Lipschitz and linear growth conditions on drift, diffusion, and jump coefficients, the authors prove that solutions inherit the Poisson stability of the coefficients and converge globally in the square-mean sense.

ABSTRACT

In this paper, we use a unified framework to study Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent and Poisson stable) solutions for semilinear stochastic differential equations driven by infinite dimensional Lévy noise with large jumps. Under suitable conditions on drift, diffusion and jump coefficients, we prove that there exist solutions which inherit the Poisson stability of coefficients. Further we show that these solutions are globally asymptotically stable in square-mean sense. Finally, we illustrate our theoretical results by several examples.

Motivation & Objective

• To establish the existence of Poisson stable solutions (including periodic, almost periodic, almost automorphic, and Levitan almost periodic) for stochastic differential equations with infinite-dimensional Lévy noise.
• To investigate whether large jumps in Lévy noise can disrupt the Poisson stability of solutions, particularly when coefficients exhibit recurrence.
• To prove that the Poisson stability of solutions is determined by the weakest recurrence property among the coefficients.
• To demonstrate global asymptotic stability in the square-mean sense for these Poisson stable solutions.
• To extend existing results on Gaussian noise-driven SDEs to the more general case of Lévy noise with jumps, including large jumps.

Proposed method

• Utilizes a unified framework based on B. A. Shcherbakov’s comparable (strongly comparable) methods for analyzing recurrence in dynamical systems.
• Applies the Lévy-Itô decomposition to separate the Lévy process into diffusive, small jump, and large jump components.
• Imposes Lipschitz and global linear growth conditions on the drift, diffusion, and jump coefficients to ensure pathwise existence and uniqueness.
• Employs semigroup theory and stochastic analysis in Hilbert spaces to analyze the abstract evolution equation driven by Lévy noise.
• Uses the square-mean norm to define stability and applies contraction mapping arguments in appropriate function spaces to prove existence and uniqueness.
• Applies the concept of distributional Poisson stability to characterize recurrence in the stochastic setting.

Experimental results

Research questions

• RQ1 Under what conditions do Poisson stable solutions exist for semilinear SDEs with infinite-dimensional Lévy noise?
• RQ2 Can solutions inherit the Poisson stability (e.g., periodicity, almost periodicity) of the coefficients despite the presence of large jumps in the Lévy noise?
• RQ3 What is the relationship between the recurrence properties of the coefficients and the resulting solution's recurrence behavior?
• RQ4 Is the Poisson stable solution globally asymptotically stable in the square-mean sense?
• RQ5 How does the weakest recurrence property among the coefficients determine the recurrence type of the solution?

Key findings

• Under suitable Lipschitz and linear growth conditions, a unique bounded solution exists that inherits the Poisson stability of the coefficients in distribution.
• The solution is globally asymptotically stable in the square-mean sense, as proven in Theorem 5.1.
• For Example 6.1, a stochastic ODE with Lévy noise admits a unique Levitan almost periodic solution in distribution if $\nu(-1,1) < \frac{25}{16}$ and $b \leq 1$.
• In Example 6.2, the stochastic heat equation with Lévy noise has a unique $\mathcal{L}^2$-bounded solution that is Levitan almost periodic in distribution and globally asymptotically stable in square-mean.
• The solution’s recurrence type is determined by the weakest recurrence property among the coefficients $f$, $g$, $F$, and $G$, as formalized in Corollary 4.7.
• All solutions with $\mathcal{L}^2$-initial values remain bounded after a sufficiently long time, as stated in Corollary 5.2.