# American Institute of Mathematical Sciences

September 2018, 8(3&4): 879-897. doi: 10.3934/mcrf.2018039

## Recurrence for switching diffusion with past dependent switching and countable state space

 1 Department of Mathematics, University of Alabama Tuscaloosa, AL 35401, USA 2 Department of Mathematics, Wayne State University, Detroit. MI 48202, USA

In honor of Jiongmin Yong on the occasion of his 60th Birthday

Received  August 2017 Revised  December 2017 Published  September 2018

Fund Project: This research was supported in part by the Air Force Office of Scientific Research under FA9550-15-1-0131. The research of D. Nguyen was also supported by the AMS-Simons Travel grant

This work continues and substantially extends our recent work on switching diffusions with the switching processes that depend on the past states and that take values in a countable state space. That is, the discrete component of the two-component process takes values in a countably infinite set and its switching rate at current time depends on the value of the continuous component involving past history. This paper focuses on recurrence, positive recurrence, and weak stabilization of such systems. In particular, the paper aims to providing more verifiable conditions on recurrence and positive recurrence and related issues. Assuming that the system is linearizable, it provides feasible conditions focusing on the coefficients of the systems for positive recurrence. Then linear feedback controls for weak stabilization are considered. Some illustrative examples are also given.

Citation: Dang H. Nguyen, George Yin. Recurrence for switching diffusion with past dependent switching and countable state space. Mathematical Control & Related Fields, 2018, 8 (3&4) : 879-897. doi: 10.3934/mcrf.2018039
##### References:
 [1] W. J. Anderson, Continuous-time Markov chains: An Applications-Oriented Approach, Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0. [2] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781. [3] M. F. Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456. [4] R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072. doi: 10.1016/j.jfa.2010.04.017. [5] R. Cont and D.-A. Fournié, Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab., 41 (2013), 109-133. doi: 10.1214/11-AOP721. [6] M. Costa, A piecewise deterministic model for a prey-predator community, Ann. Appl. Probab, 26 (2016), 3491-3530. doi: 10.1214/16-AAP1182. [7] N. T. Dieu, N. H. Du, D. H. Nguyen and G. Yin, Protection zones for survival of species in random environment, SIAM J. Appl. Math., 76 (2016), 1382-1402. doi: 10.1137/15M1032004. [8] N. T. Dieu, D. H. Nguyen, N. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM J. Appl. Dynamic Sys., 15 (2016), 1062-1084. doi: 10.1137/15M1043315. [9] N. H. Du and D. H. Nguyen, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409. doi: 10.1016/j.jde.2010.08.023. [10] N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202. doi: 10.1017/jpr.2015.18. [11] B. Dupire, Functional Itô's Calculus, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS Available at SSRN: http://ssrn.com/abstract=1435551 or http://dx.doi.org/10.2139/ssrn.1435551. [12] A. Hening, D. Nguyen and G. Yin, Stochastic population growth in spatially heterogeneous environments: The density-dependent case, J. Math. Biology, 76 (2018), 697-754. doi: 10.1007/s00285-017-1153-2. [13] S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Math. Biosc., 181 (2003), 55-83. doi: 10.1016/S0025-5564(02)00127-X. [14] R. Z. Khasminskii, C. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051. doi: 10.1016/j.spa.2006.12.001. [15] V. Kulkarni, Fluid models for single buffer systems, in Frontiers in Queueing: Models and Applications in Science and Engineering, 321–338, Ed. J. H. Dashalalow, CRC Press, 1997. [16] R. F. Luck, Evaluation of natural enemies for biological control: A behavior approach, Trends Ecol. Evol., 5 (1990), 196-199. doi: 10.1016/0169-5347(90)90210-5. [17] X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood, Chinester, 2008. doi: 10.1533/9780857099402. [18] X. Mao and C. Yuan. Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. [19] D. H. Nguyen, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101. doi: 10.1016/j.jde.2014.05.029. [20] D. H. Nguyen and G. Yin, Modeling and analysis of switching diffusion systems: Past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), 2450-2477. doi: 10.1137/16M1059357. [21] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Eqs., 262 (2017), 1192-1225. doi: 10.1016/j.jde.2016.10.005. [22] D. H. Nguyen and G. Yin, Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space, to appear in Potential Anal., (2017). [23] J. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electron. J. Probab., 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018. [24] J. Shao and F. Xi, Strong ergodicity of the regime-switching diffusion processes, Stochastic Process. Appl., 123 (2013), 3903-3918. doi: 10.1016/j.spa.2013.06.002. [25] J. Shao and F. Xi, Stability and recurrence of regime-switching diffusion processes, SIAM J. Control Optim., 52 (2014), 3496-3516. doi: 10.1137/140962905. [26] W. M. Wonham, Liapunov criteria for weak stochastic stability, J. Differential Eqs., 2 (1966), 195-207. doi: 10.1016/0022-0396(66)90043-X. [27] F. Xi and C. Zhu, On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818. doi: 10.1137/16M1087837. [28] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [29] G. Yin, H. Q. Zhang and Q. Zhang, Applications of Two-time-scale Markovian Systems, Science Press, Beijing, China, 2013. [30] G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382. doi: 10.1137/110851171. [31] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, NY, 1999. doi: 10.1007/978-1-4612-1466-3. [32] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343. [33] C. Zhu and G. Yin, On strong Feller, recurrence, and weak stabilization of regime-switching diffusions, SIAM J. Control Optim., 48 (2009), 2003-2031. doi: 10.1137/080712532. [34] X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and pth-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622. doi: 10.1137/14095251X.

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##### References:
 [1] W. J. Anderson, Continuous-time Markov chains: An Applications-Oriented Approach, Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0. [2] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781. [3] M. F. Chen, From Markov Chains to Non-equilibrium Particle Systems, World Scientific, Singapore, 2004. doi: 10.1142/9789812562456. [4] R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072. doi: 10.1016/j.jfa.2010.04.017. [5] R. Cont and D.-A. Fournié, Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab., 41 (2013), 109-133. doi: 10.1214/11-AOP721. [6] M. Costa, A piecewise deterministic model for a prey-predator community, Ann. Appl. Probab, 26 (2016), 3491-3530. doi: 10.1214/16-AAP1182. [7] N. T. Dieu, N. H. Du, D. H. Nguyen and G. Yin, Protection zones for survival of species in random environment, SIAM J. Appl. Math., 76 (2016), 1382-1402. doi: 10.1137/15M1032004. [8] N. T. Dieu, D. H. Nguyen, N. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM J. Appl. Dynamic Sys., 15 (2016), 1062-1084. doi: 10.1137/15M1043315. [9] N. H. Du and D. H. Nguyen, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409. doi: 10.1016/j.jde.2010.08.023. [10] N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 53 (2016), 187-202. doi: 10.1017/jpr.2015.18. [11] B. Dupire, Functional Itô's Calculus, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS Available at SSRN: http://ssrn.com/abstract=1435551 or http://dx.doi.org/10.2139/ssrn.1435551. [12] A. Hening, D. Nguyen and G. Yin, Stochastic population growth in spatially heterogeneous environments: The density-dependent case, J. Math. Biology, 76 (2018), 697-754. doi: 10.1007/s00285-017-1153-2. [13] S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Math. Biosc., 181 (2003), 55-83. doi: 10.1016/S0025-5564(02)00127-X. [14] R. Z. Khasminskii, C. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051. doi: 10.1016/j.spa.2006.12.001. [15] V. Kulkarni, Fluid models for single buffer systems, in Frontiers in Queueing: Models and Applications in Science and Engineering, 321–338, Ed. J. H. Dashalalow, CRC Press, 1997. [16] R. F. Luck, Evaluation of natural enemies for biological control: A behavior approach, Trends Ecol. Evol., 5 (1990), 196-199. doi: 10.1016/0169-5347(90)90210-5. [17] X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood, Chinester, 2008. doi: 10.1533/9780857099402. [18] X. Mao and C. Yuan. Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. [19] D. H. Nguyen, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101. doi: 10.1016/j.jde.2014.05.029. [20] D. H. Nguyen and G. Yin, Modeling and analysis of switching diffusion systems: Past-dependent switching with a countable state space, SIAM J. Control Optim., 54 (2016), 2450-2477. doi: 10.1137/16M1059357. [21] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Eqs., 262 (2017), 1192-1225. doi: 10.1016/j.jde.2016.10.005. [22] D. H. Nguyen and G. Yin, Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space, to appear in Potential Anal., (2017). [23] J. Shao, Criteria for transience and recurrence of regime-switching diffusion processes, Electron. J. Probab., 20 (2015), 15 pp. doi: 10.1214/EJP.v20-4018. [24] J. Shao and F. Xi, Strong ergodicity of the regime-switching diffusion processes, Stochastic Process. Appl., 123 (2013), 3903-3918. doi: 10.1016/j.spa.2013.06.002. [25] J. Shao and F. Xi, Stability and recurrence of regime-switching diffusion processes, SIAM J. Control Optim., 52 (2014), 3496-3516. doi: 10.1137/140962905. [26] W. M. Wonham, Liapunov criteria for weak stochastic stability, J. Differential Eqs., 2 (1966), 195-207. doi: 10.1016/0022-0396(66)90043-X. [27] F. Xi and C. Zhu, On Feller and strong Feller properties and exponential ergodicity of regime-switching jump diffusion processes with countable regimes, SIAM J. Control Optim., 55 (2017), 1789-1818. doi: 10.1137/16M1087837. [28] G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [29] G. Yin, H. Q. Zhang and Q. Zhang, Applications of Two-time-scale Markovian Systems, Science Press, Beijing, China, 2013. [30] G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems, SIAM J. Appl. Math., 72 (2012), 1361-1382. doi: 10.1137/110851171. [31] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, NY, 1999. doi: 10.1007/978-1-4612-1466-3. [32] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343. [33] C. Zhu and G. Yin, On strong Feller, recurrence, and weak stabilization of regime-switching diffusions, SIAM J. Control Optim., 48 (2009), 2003-2031. doi: 10.1137/080712532. [34] X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and pth-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622. doi: 10.1137/14095251X.
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