Fluctuations in a Markovian system of pairwise interacting particles
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 have 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 \textit{H. Tanaka} [Stochastic analysis, Proc. Taniguchi Int. Symp., Katata \& Kyoto/Jap. 1982, North- Holland Math. Libr. 32, 469-488 (1984; Zbl 0552.60051)].
scientific article; zbMATH DE number 4034820
scientific article