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Large deviations of Markov chains with multiple time-scales
Institution:1. Department of Electrical and Computer Engineering, The University of Texas at Austin, 2501 Speedway, EER 7.824, Austin, TX 78712, United States;2. The Harold and Inge Marcus Department of Industrial and Manufacturing Eng., College of Engineering, Pennsylvania State University, University Park, PA 16802, United States;1. Department of Mathematics, Faculty of Science, University of Ha''il, Saudi Arabia;2. Department of Mathematics, Faculty of Science, Mansoura University, Egypt;3. Institut für Mathematik, Universität Augsburg, Germany;1. Research Center of Nonlinear Science, College of Mathematics and Computer Science, Wuhan Textile University, Wuhan, 430073, PR China;2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, PR China;1. Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, Anhui 230026, China;2. Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA;3. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Abstract:For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a deterministic limit and a central limit theorem around it have already been proven in Kang and Kurtz (2013) and Kang et al. (2014). We present here a general approach to proving a large deviation principle in path space for such multi-scale Markov processes. Motivated by models arising in systems biology, we apply these large deviation results to general chemical reaction systems which exhibit multiple time-scales, and provide explicit calculations for several relevant examples.
Keywords:Large deviation principle  Multiple time-scales  Reaction networks  Markov chains  Jump diffusions  Piecewise deterministic Markov process  Comparison principle
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