A sample path large deviation principle for \(L^2\)-martingale measure processes
Subject of this paper is a class of infinite-dimensional diffusions that allow for descriptions via martingale problems involving \(L^2\)-martingale measures. The processes under consideration include the Fleming-Viot process, \({\mathcal S}^\prime(R^d)\)-valued cylindrical Brownian motion, super-Brownian motion, and a super-Brownian motion in a random environment introduced by \textit{L. Mytnik} [Ann. Probab. 24, No. 4, 1953-1978 (1996; Zbl 0874.60041)]. The main result of the paper is a sample path large deviation principle of Freidlin-Wentzell type, to which a Hamiltonian approach is taken. The occurring rate function is studied in some detail: Several equivalent representations are given, the regularity of paths with finite rate is studied, and these paths are related to a certain McKean-Vlasov equation. In the general formulation of the large deviation principle, upper and lower bounds are asymmetric in the sense that, for the lower bound, only those paths are taken into account which are sufficiently regular solutions to the McKean-Vlasov equation. In the ''non-interacting'' case of Gaussian white noise, a large deviation principle in the ordinary sense is obtained.
scientific article; zbMATH DE number 1385248