Large deviations for stochastic processes
By
Abstract
General functional large deviation
results for cadlag processes under the Skorohod topology are developed. For Markov processes,
criteria for large deviation results are given in terms convergence of
infinitesimal generators. Following an introduction
and overview, the material is presented in three parts.
Part
1 gives necessary and
sufficient conditions for exponential tightness that are analogous to
conditions for tightness in the theory of weak convergence. Analogs of the Prohorov
theorem given by Puhalskii, O'Brien and Vervaat, and de Acosta, then imply the large deviation principle,
at least along subsequences.
Representations of the rate function in terms of rate functions for the finite
dimensional distributions are extended to include situations in which the rate
function may be finite for sample paths with discontinuities.
Part
2 focuses on Markov
processes in metric spaces. For a
sequence of such processes, convergence of Fleming's logarithmically
transformed nonlinear semigroups is shown to imply
the large deviation principle in a manner analogous to the use of convergence
of linear semigroups in weak convergence. In particular cases, this convergence can be
verified using the theory of nonlinear contraction semigroups. The theory of viscosity solutions of
nonlinear equations is used to generalize earlier results on semigroup convergence, enabling the approach to cover a
wide variety of situations. The key
requirement is that a comparison principle holds. Control methods are used to
give representations of the rate functions.
Part
3 discusses methods for
verifying the comparison principle and applies the general theory to obtain a
variety of new and known results on large deviations for Markov processes. Applications include Freidlin-Wentzell
theory for nearly deterministic processes, random walks and Markov chains, Donsker-Varadhan theory for occupation measures, random
evolutions and averaging problems for stochastic systems, rescaled time-space lattice
equations, and weakly interacting particle systems. The latter results include new comparison
principles for a class of Hamilton-Jacobi equations
in Hilbert spaces and spaces of probability measures.
|
( |
|
|
( |