# 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
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