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

Dang H. Nguyen; George Yin

doi:10.3934/mcrf.2018039

Article Contents

2018, Volume 8, Issue 3&4: 879-897. Doi: 10.3934/mcrf.2018039

This issue Previous Article Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors Next Article Carleman commutator approach in logarithmic convexity for parabolic equations

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

Dang H. Nguyen^1, and
George Yin^2,

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

Received: August 2017

Revised: December 2017

Published: September 2018

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

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Abstract

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.

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Mathematics Subject Classification: 60J27, 60J60, 60K37.

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