Abstract
We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t). Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions Xt. We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (Xt, Λ(t)), and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.
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Acknowledgements
The authors would like to thank the referees and editors for many useful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant No. 12071031).
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Li, J., Xi, F. Convergence, boundedness, and ergodicity of regime-switching diffusion processes with infinite memory. Front. Math. China 16, 499–523 (2021). https://doi.org/10.1007/s11464-020-0863-8
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DOI: https://doi.org/10.1007/s11464-020-0863-8
Keywords
- Regime-switching diffusion process
- infinite memory
- convergence
- boundedness
- Feller property
- invariant measure
- Wasserstein distance