Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling |
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| Institution: | 1. School of Finance and Statistics, East China Normal University, Shanghai 200241, China;2. Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia;3. School of Risk and Actuarial Studies and CEPAR, Australian School of Business, University of New South Wales, Sydney, NSW 2052, Australia;1. Department of Bioengineering, Rice University, 6100 Main Street, Houston, TX 77005, United States;2. Department of Biosciences, Rice University, 6100 Main Street, Houston, TX 77005, United States;3. Center for Theoretical Biophysics, Rice University, 6100 Main Street, Houston, TX 77005, United States |
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| Abstract: | We investigate a piecewise-deterministic Markov process with a Polish state space, whose deterministic behaviour between random jumps is governed by a finite number of semiflows. We provide tractable conditions ensuring a form of exponential ergodicity and the strong law of large numbers for the chain given by the post-jump locations. Further, we establish a one-to-one correspondence between invariant measures of the chain and those of the continuous-time process. These results enable us to derive the strong law of large numbers for the latter. The studied dynamical system is inspired by certain models of gene expression, which are also discussed here. |
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| Keywords: | Markov process Invariant measure Exponential ergodicity Asymptotic stability The strong law of large numbers Gene expression |
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