Multilevel large deviations and interacting diffusions
Let \((\xi^ N)\) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For each \(M\) and each \(N\), let \(\Xi^{M,N}\) denote the empirical measure associated with \(M\) independent copies of \(\xi^ N\). As a main result, we show that \((\Xi^{M,N})\) also satisfies the large deviation principle as \(M,N \to \infty\). We derive several representations of the associated rate function. These results are then applied to empirical measure processes \(\Xi^{M,N} (t)=M^{-1} \sum^ M_{i=1} \delta_{\xi^ N_ i(t)}\), \(0 \leq t \leq T\), where \((\xi^ N_ 1(t), \dots, \xi^ N_ M(t))\) is a system of weakly interacting diffusions with noise intensity \(1/N\). This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models [Stochastics 20, 247- 308 (1987; Zbl 0613.60021) and Stochastic calculus in applications, Proc. Symp. Cambridge/UK 1987, Pitman Res. Notes Math. Ser. 197, 29-54 (1988; Zbl 0668.60067)].
scientific article; zbMATH DE number 569026
scientific article