# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021309
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## Exponential ergodicity for regime-switching diffusion processes in total variation norm

 1 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Fubao Xi

Received  July 2021 Revised  November 2021 Early access January 2022

Fund Project: The research was supported by the National Natural Science Foundation of China (12071031)

We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.

Citation: Jun Li, Fubao Xi. Exponential ergodicity for regime-switching diffusion processes in total variation norm. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021309
##### References:
 [1] J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, Nonlinear Anal. Hybrid Syst., 38 (2020), 100914, 18 pp. doi: 10.1016/j.nahs.2020.100914. [2] J. Bao, J. Shao and C. Yuan, Approximation of invariant measures for regime-switching diffusions, Potential Anal., 44 (2016), 707-727.  doi: 10.1007/s11118-015-9526-x. [3] J.-B. Bardet, H. Guérin and F. Malrieu, Long time behavior of diffusions with Markov switching, ALEA Lat. Am. J. Probab. Math. Stat., 7 (2010), 151-170. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262. [5] M.-F. Chen, From Markov Chains to non-Equillibrium Particle Systems, World Scientific Publishing Co. Pte. Ltd, Singapore, 2004. doi: 10.1142/9789812562456. [6] M.-F. Chen and S. Li, Coupling methods for multidimensional diffusion processes, Ann. Probab., 17 (1989), 151-177. [7] B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577. [8] B. de Saporta and J.-F. Yao, Tail of linear diffusion with Markov switching, Ann. Appl. Probab., 15 (2005), 992-1018.  doi: 10.1214/105051604000000828. [9] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New York, 1964. [10] T. Hou and J. Shao, Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching, Sci. China Math., 63 (2020), 1169-1180.  doi: 10.1007/s11425-017-9392-5. [11] Y. Hu, D. Nualart, X. Sun and Y. Xie, Smoothness of density for stochastic differential equations with Markovian switching, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3615-3631.  doi: 10.3934/dcdsb.2018307. [12] H. Ji and F. Xi, Stationary distribution of stochastic population dynamics in state-dependent random environments, Systems Control Lett., 144 (2020), 104774, 8 pp. doi: 10.1016/j.sysconle.2020.104774. [13] R. Khasminskii, Stochastic Stability of Differential Equations, 2$^{nd}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-3-642-23280-0. [14] N. V. Krylov, Introduction to the Theory of Diffusion Processes, American Math. Soc., Providence, RI, 2002. doi: 10.1090/gsm/043. [15] A. Kulik, Ergodic Behavior of Markov Processes, De Gruyter, Berlin, 2018. [16] X. Ma and F. Xi, Large deviations for empirical measures of switching diffusion processes with small parameters, Front. Math. China, 10 (2015), 949-963.  doi: 10.1007/s11464-015-0486-7. [17] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473. [18] D. H. Nguyen, G. Yin and C. Zhu, Certain properties related to well posedness of switching diffusions, Stoch. Process. Appl., 127 (2017), 3135-3158.  doi: 10.1016/j.spa.2017.02.004. [19] R. Pinsky and M. Scheutzow, Some remarks and examples concerning the transience and recurrence of random diffusions, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 519-536. [20] G. O. Roberts and J. S. Rosenthal, Quantitative bounds for convergence rates of continuous time Markov processes, Electron. J. Probab., 1 (1996), 21 pp. doi: 10.1214/EJP.v1-9. [21] J. Shao, Ergodicity of one-dimensional regime-switching diffusion processes, Sci. China Math., 57 (2014), 2407-2414.  doi: 10.1007/s11425-014-4853-8. [22] J. Shao, Ergodicity of regime-switching diffusions in Wasserstein distances, Stoch. Process. Appl., 125 (2015), 739-758.  doi: 10.1016/j.spa.2014.10.007. [23] J. Shao, Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions, SIAM J. Contr. Optim., 56 (2018), 3215-3238.  doi: 10.1137/18M116678X. [24] X. Sun and Y. Xie, Smooth densities for SDEs driven by subordinated Brownian motion with Markovian switching, Front. Math. China, 13 (2018), 1447-1467.  doi: 10.1007/s11464-018-0735-7. [25] A. Yu. Veretennikov, Bounds for the mixing rate in the theory of stochastic equations, Theory Probab. Appl., 32 (1988), 273-281. [26] A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stoch. Process. Appl., 70 (1997), 115-127.  doi: 10.1016/S0304-4149(97)00056-2. [27] F. Xi, Invariant measures for a random evolution equation with small perturbations, Acta Math. Appl. Sin. Engl. Ser., 17 (2001), 631-642.  doi: 10.1007/s101140100127. [28] F. Xi, Stability of a random diffusion with nonlinear drift, Stat. Probab. Letters, 68 (2004), 273-286.  doi: 10.1016/j.spl.2004.03.010. [29] F. Xi, Feller property and exponential ergodicity of diffusion processes with state-dependent switching, Sci. China Ser. A, 51 (2008), 329-342.  doi: 10.1007/s11425-007-0147-8. [30] F. Xi and G. Yin, Asymptotic properties of a mean-field model with a continuous-state-dependent switching process, J. Appl. Probab., 46 (2009), 221-243.  doi: 10.1239/jap/1238592126. [31] F. Xi and G. Yin, Jump-diffusions with state-dependent switching: Existence and uniqueness, Feller property, linearization, and uniform ergodicity, Sci. China Math., 54 (2011), 2651-2667.  doi: 10.1007/s11425-011-4281-y. [32] F. Xi and G. Yin, Stochastic Liénard equations with state-dependent switching, Acta Math. Appl. Sin. Engl. Ser., 31 (2015), 893-908.  doi: 10.1007/s10255-015-0538-5. [33] F. Xi, G. Yin and C. Zhu, Regime-switching jump diffusions with non-Lipschitz coefficients and countably many switching states: Existence and uniqueness, Feller, and strong Feller properties, in Modeling, Stochastic Control, Optimization, and Applications (eds. G. Yin and Q. Zhang), Springer, (2019), 571–599. [34] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [35] C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stoch. Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2. [36] Z. Zhang, H. Yang, J. Tong and L. Hu, Necessary and sufficient conditions for ergodicity of CIR type SDEs with Markov switching, Stoch. Dyn., 19 (2019), 1950023, 26 pp. doi: 10.1142/S0219493719500230.

show all references

##### References:
 [1] J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, Nonlinear Anal. Hybrid Syst., 38 (2020), 100914, 18 pp. doi: 10.1016/j.nahs.2020.100914. [2] J. Bao, J. Shao and C. Yuan, Approximation of invariant measures for regime-switching diffusions, Potential Anal., 44 (2016), 707-727.  doi: 10.1007/s11118-015-9526-x. [3] J.-B. Bardet, H. Guérin and F. Malrieu, Long time behavior of diffusions with Markov switching, ALEA Lat. Am. J. Probab. Math. Stat., 7 (2010), 151-170. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262. [5] M.-F. Chen, From Markov Chains to non-Equillibrium Particle Systems, World Scientific Publishing Co. Pte. Ltd, Singapore, 2004. doi: 10.1142/9789812562456. [6] M.-F. Chen and S. Li, Coupling methods for multidimensional diffusion processes, Ann. Probab., 17 (1989), 151-177. [7] B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577. [8] B. de Saporta and J.-F. Yao, Tail of linear diffusion with Markov switching, Ann. Appl. Probab., 15 (2005), 992-1018.  doi: 10.1214/105051604000000828. [9] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New York, 1964. [10] T. Hou and J. Shao, Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching, Sci. China Math., 63 (2020), 1169-1180.  doi: 10.1007/s11425-017-9392-5. [11] Y. Hu, D. Nualart, X. Sun and Y. Xie, Smoothness of density for stochastic differential equations with Markovian switching, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3615-3631.  doi: 10.3934/dcdsb.2018307. [12] H. Ji and F. Xi, Stationary distribution of stochastic population dynamics in state-dependent random environments, Systems Control Lett., 144 (2020), 104774, 8 pp. doi: 10.1016/j.sysconle.2020.104774. [13] R. Khasminskii, Stochastic Stability of Differential Equations, 2$^{nd}$ edition, Springer, Berlin, 2012. doi: 10.1007/978-3-642-23280-0. [14] N. V. Krylov, Introduction to the Theory of Diffusion Processes, American Math. Soc., Providence, RI, 2002. doi: 10.1090/gsm/043. [15] A. Kulik, Ergodic Behavior of Markov Processes, De Gruyter, Berlin, 2018. [16] X. Ma and F. Xi, Large deviations for empirical measures of switching diffusion processes with small parameters, Front. Math. China, 10 (2015), 949-963.  doi: 10.1007/s11464-015-0486-7. [17] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473. [18] D. H. Nguyen, G. Yin and C. Zhu, Certain properties related to well posedness of switching diffusions, Stoch. Process. Appl., 127 (2017), 3135-3158.  doi: 10.1016/j.spa.2017.02.004. [19] R. Pinsky and M. Scheutzow, Some remarks and examples concerning the transience and recurrence of random diffusions, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 519-536. [20] G. O. Roberts and J. S. Rosenthal, Quantitative bounds for convergence rates of continuous time Markov processes, Electron. J. Probab., 1 (1996), 21 pp. doi: 10.1214/EJP.v1-9. [21] J. Shao, Ergodicity of one-dimensional regime-switching diffusion processes, Sci. China Math., 57 (2014), 2407-2414.  doi: 10.1007/s11425-014-4853-8. [22] J. Shao, Ergodicity of regime-switching diffusions in Wasserstein distances, Stoch. Process. Appl., 125 (2015), 739-758.  doi: 10.1016/j.spa.2014.10.007. [23] J. Shao, Invariant measures and Euler-Maruyama's approximations of state-dependent regime-switching diffusions, SIAM J. Contr. Optim., 56 (2018), 3215-3238.  doi: 10.1137/18M116678X. [24] X. Sun and Y. Xie, Smooth densities for SDEs driven by subordinated Brownian motion with Markovian switching, Front. Math. China, 13 (2018), 1447-1467.  doi: 10.1007/s11464-018-0735-7. [25] A. Yu. Veretennikov, Bounds for the mixing rate in the theory of stochastic equations, Theory Probab. Appl., 32 (1988), 273-281. [26] A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stoch. Process. Appl., 70 (1997), 115-127.  doi: 10.1016/S0304-4149(97)00056-2. [27] F. Xi, Invariant measures for a random evolution equation with small perturbations, Acta Math. Appl. Sin. Engl. Ser., 17 (2001), 631-642.  doi: 10.1007/s101140100127. [28] F. Xi, Stability of a random diffusion with nonlinear drift, Stat. Probab. Letters, 68 (2004), 273-286.  doi: 10.1016/j.spl.2004.03.010. [29] F. Xi, Feller property and exponential ergodicity of diffusion processes with state-dependent switching, Sci. China Ser. A, 51 (2008), 329-342.  doi: 10.1007/s11425-007-0147-8. [30] F. Xi and G. Yin, Asymptotic properties of a mean-field model with a continuous-state-dependent switching process, J. Appl. Probab., 46 (2009), 221-243.  doi: 10.1239/jap/1238592126. [31] F. Xi and G. Yin, Jump-diffusions with state-dependent switching: Existence and uniqueness, Feller property, linearization, and uniform ergodicity, Sci. China Math., 54 (2011), 2651-2667.  doi: 10.1007/s11425-011-4281-y. [32] F. Xi and G. Yin, Stochastic Liénard equations with state-dependent switching, Acta Math. Appl. Sin. Engl. Ser., 31 (2015), 893-908.  doi: 10.1007/s10255-015-0538-5. [33] F. Xi, G. Yin and C. Zhu, Regime-switching jump diffusions with non-Lipschitz coefficients and countably many switching states: Existence and uniqueness, Feller, and strong Feller properties, in Modeling, Stochastic Control, Optimization, and Applications (eds. G. Yin and Q. Zhang), Springer, (2019), 571–599. [34] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [35] C. Yuan and X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stoch. Process. Appl., 103 (2003), 277-291.  doi: 10.1016/S0304-4149(02)00230-2. [36] Z. Zhang, H. Yang, J. Tong and L. Hu, Necessary and sufficient conditions for ergodicity of CIR type SDEs with Markov switching, Stoch. Dyn., 19 (2019), 1950023, 26 pp. doi: 10.1142/S0219493719500230.
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