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Asymptotic Properties of a Mean-Field Model with a Continuous-State-Dependent Switching Process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Fubao Xi  and
G. Yin
Fubao Xi*
Affiliation:
Beijing Institute of Technology
G. Yin*
Affiliation:
Wayne State University
*
∗Postal address: Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, P. R. China. Email address: xifb@bit.edu.cn
∗∗Postal address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: gyin@math.wayne.edu
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Abstract

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This work is concerned with a class of mean-field models given by a switching diffusion with a continuous-state-dependent switching process. Focusing on asymptotic properties, the regularity or nonexplosiveness, Feller continuity, and strong Feller continuity are established by means of introducing certain auxiliary processes and by making use of the truncations. Based on these results, exponential ergodicity is obtained under the Foster–Lyapunov drift conditions. By virtue of the coupling methods, the strong ergodicity or uniform ergodicity in the sense of convergence in the variation norm is established for the mean-field model with a Markovian switching process. Besides this, several examples are presented for demonstration and illustration.

Keywords

Mean-field modelcontinuous-state-dependent switchingMarkovian switchingFeller continuitystrong Fellercontinuityexponential ergodicitystrong ergodicity

MSC classification

Secondary: 60J60: Diffusion processes 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 221 - 243
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Supported in part by the National Natural Science Foundation of China under grant number 10671037.

Supported in part by the National Science Foundation under grant DMS-0603287 and in part by the National Security Agency under grant MSPF-068-029.

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