Large deviations for empirical process of mean-field interacting particle system with unbounded jumps
A particle system with mean-field interaction is considered. It is described as an \(N\)-dimensional Markov process. It is known that, for any finite time interval, the corresponding empirical process satisfies a law of large number as \(N \to \infty\) and that the limiting deterministic dynamics are characterized as a weak solution of a nonlinear master equation. The nonlinear master equation was proposed by \textit{G. Nicolis} and \textit{I. Prigogine} [``Self-organization in nonequilibrium systems. From dissipative structures to order through fluctuations'' (1977; Zbl 0363.93005)]. \textit{S. Feng} and \textit{X. Zheng} [Stochastic Processes Appl. 43, No. 1, 65-84 (1992; Zbl 0762.60075)] proved the existence and uniqueness of the solution to the nonlinear master equation. For the second Schögl model, the existence of at least three equilibrium states is proved. The motivation of the paper is to investigate the long-time behavior such as tunnelling and metastability of the \(N\)-particle systems. Consider the case when the nonlinear master equation has more than one equilibrium state. From the law of large numbers we can see that if \(N\) is large but finite, then the empirical process will normally follow the path of the dynamical system underlying the nonlinear master equation, that is attracted by one of the equilibrium states. As time goes on, the particle system will make small fluctuations near the equilibrium states. However, because of ergodicity, the transition from the neighborhood of one equilibrium state to another may eventually occur via large deviations. The purpose of the paper is to study the large deviations of the empirical process and to find a nice representation for the action functional. The special feature of the model discussed in the paper is that it is not only a pure jump Markov process with mean-field interaction, but also with unbounded jumps. Similar problems have been discussed by \textit{M. I. Frejdlin} and \textit{A. D. Venttsel'} [``Random perturbations of dynamical systems'' (1984; Zbl 0522.60055)]; \textit{D. A. Dawson} and \textit{J. Gärtner} [Stochastic 20, 247-308 (1987; Zbl 0613.60021)]; \textit{F. Comets} [Ann. Inst. H. Poincaré Probab. Stat. 23, 135-178 (1987; Zbl 0633.60110)]; \textit{M. Sugiura} [Preprint (1990)]\ and \textit{C. Léonard} [Preprint (1990)].
scientific article; zbMATH DE number 828378