Ergodicity for regime-switching neutral stochastic functional differential equations with infinite delay

Abstract: This work focuses on a class of regime-switching neutral stochastic functional differential equations (RNSFDEs) with infinite delay, in which the switching component can possess finite or countably infinite many states. To ensure the well-posedness of the underlying process, we first investigate the well-posedness for NSFDEs without Markovian switching under dissipativity conditions, and obtain the desired result by Skorohod's representation. By utilizing the moment estimate of exponential functionals of the switching component, we derive the exponential ergodicity in Wasserstein distance for RNSFDEs with a finite state space using the coupling method. To address the difficulty posed by the infinite state space, we obtain the same exponential ergodicity by applying the finite partition method along with Lyapunov functions and M-matrix theory.

• The paper establishes explicit exponential ergodicity in Wasserstein distance for RNSFDEs with infinite delay under dissipativity and boundedness conditions.
• The paper employs coupling constructions and Lyapunov-M-matrix techniques to ensure pathwise uniqueness and uniform L^p-moment bounds in both finite and infinite regime spaces.
• The paper demonstrates strong numerical and theoretical implications, paving the way for convergence analysis of Euler-type schemes and further research on state-dependent switching.

Exponential Ergodicity of Regime-Switching Neutral Stochastic Functional Differential Equations with Infinite Delay

Introduction and Motivation

This work establishes sharp results on the exponential ergodicity of regime-switching neutral stochastic functional differential equations (RNSFDEs) with infinite delay, where the switching component can have either a finite or countably infinite state space (2604.10244). Such systems are mathematically intricate because solutions depend on the entire infinite past and the switching mechanism, which introduces additional complexity via Markovian regime changes. This combination is central to modeling memory in stochastic dynamical systems subject to abrupt regime changes and is relevant in fields ranging from control theory to mathematical finance.

A distinguishing aspect of RNSFDEs compared to classical SFDEs is the presence of neutral terms, where the derivative of the process explicitly depends on delayed arguments. The regime-switching further incorporates Markov-modulated coefficients, substantially complicating the study of long-term behavior. Existing work on ergodicity for regime-switching SFDEs is limited, especially for infinite-dimensional state spaces and when neutral terms are present.

Mathematical Formulation

The paper considers the RNSFDE on the Banach space $C_r$ of continuous functions from $(-\infty,0]$ to $\mathbb{R}^d$, with the weighted supremum norm. The dynamics are given by

$$\mathrm{d} \big\{X(t) - G(X_t, \Lambda(t))\big\} = b(X_t, \Lambda(t))\,\mathrm{d}t + \sigma(X_t, \Lambda(t))\,\mathrm{d}W(t)$$

Here, $X_t$ denotes the segment process, $G$ is a neutral operator, $b$ and $\sigma$ are Borel measurable drift and diffusion coefficients, and $W(t)$ is a $d$-dimensional Brownian motion. The regime-switching process $(-\infty,0]$0 is a continuous-time Markov chain on either a finite or countably infinite state space with generator $(-\infty,0]$1.

The neutral functional differential structure, infinite memory, and regime-switching together yield a non-Markov process (in the solution variable) and, crucially, bring forth significant technical barriers; for example, standard functional inequalities or the strong Feller property do not hold in this setting.

Well-Posedness and Existence of Strong Solutions

The authors first address the well-posedness by analyzing the RNSFDE without switching, under dissipativity and Lipschitz-type conditions for the coefficients. A truncation argument and Skorokhod's representation are employed, along with careful $(-\infty,0]$2-estimates tailored to the infinite delay and neutral term, yielding existence and pathwise uniqueness of global solutions in $(-\infty,0]$3 for each regime.

For the full system with regime-switching, existence and uniqueness are extended by leveraging a Poisson random measure representation of the Markov chain [nguyen2016modeling], which provides a unified probability space supporting both the Brownian motion and the switching. The result holds for both finite and infinite state spaces, provided certain dissipativity and boundedness conditions (including uniform control on the rate matrix rows for the infinite state scenario).

Exponential Ergodicity in Wasserstein Distance

Finite State Space Case

For finite state spaces, the analysis is centered on exponential ergodicity in the $(-\infty,0]$4 Wasserstein metric. Key steps are:

• Uniform $(-\infty,0]$5-moment estimates for both the solution and its segment process, using the properties of the Markovian switching and the dissipative structure.
• Coupling construction: The process is coupled in both the continuous (functional) and discrete (switching) components. The Markov chain admits an explicit basic coupling where the two chains attempt to synchronize, and after coupling, their paths coincide.
• Sharp contractivity estimates: By estimating the difference between solutions started from different initial conditions and using the exponential mixing of the finite Markov chain, the contraction of the Wasserstein distance is demonstrated.
• The existence and uniqueness of an invariant measure $(-\infty,0]$6 is shown, with exponential convergence in $(-\infty,0]$7 from arbitrary initial distributions.

The Lyapunov method and $(-\infty,0]$8-matrix theory play a crucial role in obtaining the uniform moment bounds and contractivity necessary for the coupling argument to yield exponential rates.

Infinite State Space Case

For countably infinite state spaces, the main challenge is the lack of uniform minorization and irreducibility properties for the infinite Markov chain. The methodology involves:

• Finite partition technique [SHAO2015SPA]: The infinite state space is partitioned according to the drift's dissipativity, and a reduced finite-dimensional rate matrix $(-\infty,0]$9 is constructed to control the system's behavior.
• Lyapunov-type functions indexed by the partition (constructed via $\mathbb{R}^d$0-matrix theory) and generalized Foster-Lyapunov conditions are shown to hold.
• Under further technical assumptions—control on the total rates, uniform dissipativity, boundedness of coefficients—the existence and uniqueness of an invariant measure and exponential ergodicity in Wasserstein distance are again established.

Notably, for both finite and infinite switching, explicit rates of convergence are obtained, and the existence of strong solutions with the Feller property is ensured under the stated conditions.

Numerical and Theoretical Implications

Numerical implications: The results provide theoretical justification for the use of explicit or implicit Euler-type schemes for RNSFDEs, including quantification of their long-term behavior. The uniform $\mathbb{R}^d$1-bounds and contractivity pave the way for convergence analysis of numerical methods.

Theoretical implications: The work establishes robust conditions for exponential ergodicity in infinite-dimensional Markov systems with switching, strengthening and generalizing previous results in the literature for switching diffusion processes [li2021convergence, bao2020ergodicity, SHAO2015SPA]. The Lyapunov-M-matrix approach adopted here is notable for its applicability to high-dimensional, infinite-memory, and multifaceted switching systems.

The proofs and constructions also clarify that the strong Feller property is generally absent in this class, emphasizing the importance of Wasserstein metrics and pathwise coupling in the analysis of ergodicity for non-Markovian, infinite-dimensional dynamics.

Bold/Contradictory Claims and Strong Numerical Results

The paper establishes exponential ergodicity in the Wasserstein distance under fairly general dissipativity and boundedness conditions, for both neutral and non-neutral SFDEs with arbitrarily large (even infinite) Markovian regime spaces. This includes explicit rates of contraction and the uniqueness of invariant measures. The authors demonstrate that coupling and Lyapunov approaches are effective even in the absence of strong Feller or uniform minorization properties, contradicting a common limitation in SPDE theory.

Strong numerical (analytic) results include:

• Explicit Wasserstein exponential rates are derived (see Theorems in Sections 4 and 5).
• Uniform $\mathbb{R}^d$2-moment bounds, uniform in initial data and time.
• Convergence results extend to both segment and solution processes, not just the finite-dimensional marginals.

Future Directions

The authors highlight several directions for future investigation:

• Extension to state- or history-dependent switching, where transition rates depend functionally on the entire infinite history of the system state. This complicates both the probabilistic representation of the switching process and the coupling argument; existing techniques do not directly apply.
• Relaxation of dissipativity/contractivity assumptions for further generalization.
• Numerical analysis and ergodicity-preserving discretization, particularly for neutral and regime-switching systems.
• Generalization to more complex hybrid systems, including jump diffusions and systems driven by Lévy processes.
• Analysis of rare events and large deviations in the infinite-delay, regime-switching context.

Conclusion

This work significantly advances the theory of long-time behavior for regime-switching neutral SFDEs with infinite delay, establishing exponential ergodicity in Wasserstein distance for both finite and infinite regime spaces. The paper's approach blends advanced stochastic analysis, Lyapunov-M-matrix techniques, and coupling methods, setting a solid foundation for further exploration of complex memory-dependent and regime-switching stochastic systems. The methodology has potential impact on the study of stochastic control, filtering, and ergodic theory for infinite-dimensional and path-dependent processes.

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Reference:

• "Ergodicity for regime-switching neutral stochastic functional differential equations with infinite delay" (2604.10244)