On large deviations for ensembles of distributions

D. A. Khrychev

Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1671-1690

Résumé

The paper is concerned with the large deviations problem in the Freidlin-Wentzell formulation without the assumption of the uniqueness of the solution to the equation involving white noise. In other words, it is assumed that for each $\varepsilon>0$ the nonempty set $\mathscr P_\varepsilon$ of weak solutions is not necessarily a singleton. Analogues of a number of concepts in the theory of large deviations are introduced for the set $\{\mathscr P_\varepsilon,\,\varepsilon>0\}$, hereafter referred to as an ensemble of distributions. The ensembles of weak solutions of an $n$-dimensional stochastic Navier-Stokes system and stochastic wave equation with power-law nonlinearity are shown to be uniformly exponentially tight. An idempotent Wiener process in a Hilbert space and idempotent partial differential equations are defined. The accumulation points in the sense of large deviations of the ensembles in question are shown to be weak solutions of the corresponding idempotent equations. Bibliography: 14 titles.

Détail

Keywords: large deviations, $n$-dimensional Navier-Stokes system, nonlinear wave equation.