Summary
Analogously to the Kac's master equation approach to the spatially homogeneous Boltzmann equation we introduce a system of Markov processes of many particles moving on a countable set with pairwise interaction, and investigate the fluctuation around McKean's non-linear limit process. Our model possibly admits simultaneous jumps of two particles, which make impossible both such characterizations of the fluctuation in the limit and techniques (based on the Cameron-Martin formula) as has previously been obtained or used for diffusion models and for McKean's two-speed gas model. We obtain a new description of the variance functional for the fluctuation, and, by applying it in the case of no simultaneous jumps, give a new derivation of a formula of H. Tanaka [7].
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Uchiyama, K. Fluctuations in a Markovian system of pairwise interacting particles. Probab. Th. Rel. Fields 79, 289–302 (1988). https://doi.org/10.1007/BF00320923
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DOI: https://doi.org/10.1007/BF00320923
Keywords
- Stochastic Process
- Probability Theory
- Markov Process
- Diffusion Model
- Boltzmann Equation