On meeting and merging of stochastic flow of non-homogeneous Markov and semi-Markov dynamics

Abstract

We consider a general class of semi-Markov processes on countable state-space, with a differentiable kernel such that the embedded Markov chain may not be homogeneous. Using an SDE representation of the process, we study the associated flow by investigating meetings and merging related events for solutions, starting with different initial conditions. Some sufficient conditions for almost sure meeting and merging are also obtained. Many examples are considered to clarify the intricacies and implications of this study.

Keywords:

• Meeting in next transition
• merging probability
• non-homogeneous emi-Markov rocesses
• Poisson random measure
• Stochastic flow

Acknowledgments

We acknowledge the anonymous referee for giving useful suggestions which improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author would like to acknowledge the support from DST/INTDAAD/P-12/2020, as well as 02011/32/2023NBHM(R.P)/R & D II/13347. The research of the second author is supported by the Mathematical Research Impact Centric Support (MATRICS) grant, File No: MTR/2020/000350, by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India. The third author would like to acknowledge the support from CSIR SRF. The support and the resources provided by ‘PARAM Brahma Facility’ under the National Supercomputing Mission, Government of India at the Indian Institute of Science Education and Research (IISER) Pune are gratefully acknowledged.