Exponential convergence for a system of conuclear space-valued diffusions with mean-field interaction (Q1288984)

The author considers an interacting system of diffusion processes \(\{X_{j}^{n}(t):t\geq 0\}\), \(j=1,\ldots,n\), taking values in a conuclear space \(\Phi'\) and studies the rate of convergence of the empirical process. The convergence to a deterministic measure-valued process which is the unique solution of a nonlinear differential equation in the space of probabilities on \(\Phi'\) was proved by the author in a previous work. The rate of convergence was studied by \textit{T. Shiga} and \textit{H. Tanaka} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 69, 439-459 (1985; Zbl 0607.60095)] when \(\Phi'\) is of finite dimension and by \textit{G. Kallianpur} and \textit{I. Mitoma} [Can. J. Math. 44, No. 3, 524-552 (1992; Zbl 0764.46040)] for the infinite-dimensional case. The problem of the exponential convergence is studied for systems with mean-field interaction on both diffusion and drift terms [see also \textit{D. A. Dawson} and \textit{J. Gärtner}, Stochastics 20, 247-308 (1987; Zbl 0613.60021)] by showing that the sequence of laws of the empirical processes is exponentially tight. For this the author uses the fact that the empirical process satisfies a measure-valued stochastic differential equation.