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.
From: Chao Zhu [view email]
[v1]
Thu, 7 Aug 2025 03:11:47 UTC (41 KB)
[v2]
Sat, 12 Sep 2026 15:34:22 UTC (37 KB)
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