Occupation measure functionals in merging phase space

Abstract

This paper deals with the averaging and diffusion approximation for occupation time of general phase space semi-Markov processes under split with simple and double merging in a series scheme. These results seem to be particularly interesting since the variance coefficient is obtained in a simple form. The method of proof of weak convergence results here is based on the construction of the compensating operator of the extended Markov renewal processes and on the solution of the singular perturbation problem of this operator acting on test functions.

Keywords:

• occupation measure
• semi-Markov process
• averaging principle
• diffusion approximation
• singular perturbation
• random evolution

AMS Subject Classification::

• 60K15
• 60K37
• 60J55
• 60B10
• 82D30

Acknowledgements

We are grateful to Prof. Yu. Kondratiev for fruitful and stimulating discussions concerning the subject of this paper. The financial support of the DFG through the SFB 701 (Bielefeld University) and German-Ukrainian Projects 436 UKR 113/70 and 436 UKR 113/80 is gratefully acknowledged.