Large deviations of cascade processes on graphs

Large deviations of cascade processes on graphs.

Simple models of irreversible dynamical processes such as bootstrap percolation have been successfully applied to describe cascade processes in a large variety of different contexts. However, the problem of analyzing nontypical trajectories, which can be crucial for the understanding of out-of-equilibrium phenomena, is still considered to be intractable in most cases. Here we introduce an effic...

Nonequilibrium Markov Processes Conditioned on Large Deviations

Large Deviations for Stochastic Processes

The purpose of these lectures is to introduce you to the basics of large deviation theory. The emphasis will be on the use of compactness ideas (more extensive results are in Puhalskii [13]). Other approaches to large deviation theory are considered in Dembo and Zeitouni [3], den Hollander [4], Deuschel and Stroock [6], Dupuis and Ellis [7], Freidlin and Wentzell [9], Kallenberg [12], Shwartz a...

Large deviations for Wishart processes

Let X(δ) be a Wishart process of dimension δ, with values in the set of positive matrices of size m. We are interested in the large deviations for a family of matrix-valued processes {δ−1X t , t ≤ 1} as δ tends to infinity. The process X(δ) is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential mart...

Large Deviations for Stochastic Processes

The notes are devoted to results on large deviations for sequences of Markov processes following closely the book by Feng and Kurtz ([FK06]). We outline how convergence of Fleming’s nonlinear semigroups (logarithmically transformed nonlinear semigroups) implies large deviation principles analogous to the use of convergence of linear semigroups in weak convergence. The latter method is based on ...