Result: On large deviations for a multidimensional random process with slow and fast diffusities

The author proves a large deviations principle for a family of multidimensional diffusion processes with the diffusion matrix depending on \((x,y)\in\mathbb R^{r}\times[-b,b]\). The main motivation is the asymptotic study of the solution \(u^{\varepsilon}(t,x,y)\) (concentration of particles) of a nonlinear second-order initial-boundary problem of KPP-type. The second-order operator generates a fast diffusion (order \(1/\varepsilon\)) in \(y\)-direction and a slow diffusion (order \(\varepsilon\)) in \(x\)-direction. The nonlinear term governs the multiplication (killing) of particles. The relation between \(u^{\varepsilon}\) and the diffusion process is given by the Feynman-Kac formula. The one-dimensional case has been studied by the author [Stochastics Stochastics Rep. 52, No. 1-2, 43-80 and 81-105 (1995; Zbl 0864.60050 and Zbl 0864.60051)]. The multidimensional case has also been considered by \textit{M. I. Freidlin} and \textit{T.-Y. Lee} [Probab. Theory Relat. Fields 105, No. 2, 227-254 (1996; Zbl 0847.60020) and ibid. 106, No. 1, 39-70 (1996; Zbl 0855.60075)]. The method of proof consists in using \textit{M. D. Donsker} and \textit{S. R. S. Varadhan}'s approach [in: Funct. Integr. Appl., Proc. int. Conf., London 1974, 15-33 (1975; Zbl 0333.60078)] for guessing what should be the action functional and then proving that it is the desired action functional [see also \textit{A. D. Venttsel'}, Theory Probab. Appl. 21(1976), 227-242 (1977), translation from Teor. Veroyatn. Primen. 21, 235-252 (1976; Zbl 0361.60005) and ibid. 21(1976), 499-512 (1977) resp. ibid. 21, 512-526 (1976; Zbl 0361.60006)].