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arXiv · 2508.04997

Switching Diffusion Systems with Past-Dependent Switching and Countable State Space: Successful Couplings and Strong Ergodicity

Fubao Xi·Yafei Zhai·Chao Zhu
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Abstract

Motivated by applications in biology, ecology, and finance, this paper investigates a class of switching diffusion systems where the discrete component takes values in a countable state space with transition rates dependent on the history of the continuous component. While history-dependent transition rates provide greater modeling flexibility for complex systems, they also introduce substantial analytical challenges due to the non-Markovian dynamics. In particular, successful coupling differs fundamentally from the delay-free setting: the meeting of two coupled processes does not immediately imply successful coupling, since the processes may separate again after coalescence as a result of their history-dependent switching. Consequently, successful coupling requires the two processes to remain identical for a sufficiently long period after their initial meeting. Under suitable conditions, we construct such a coupling, establish stability of the associated segment process in the total variation norm, and derive its strong ergodicity. Finally, we illustrate our theoretical results through an $N$-body mean-field model with past-dependent switching on a countable state space.

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Fubao Xi, Yafei Zhai, Chao Zhu. 2026-09-12. Switching Diffusion Systems with Past-Dependent Switching and Countable State Space: Successful Couplings and Strong Ergodicity. https://arxiv.org/abs/2508.04997

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