Large Deviation Theory for Stochastic Diierence Equations
نویسنده
چکیده
The probability density for the solution yn of a stochastic di erence equation is considered. Following Knessl, Matkowsky, Schuss, and Tier [1] it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a di erence approximation to the Helmholtz equation in a random medium. A large deviation result is obtained for the probability density of the decay rate of a solution of this equation. Both the exponent and the pre-exponential factor are determined.
منابع مشابه
Large Deviations for the Shell Model of Turbulence Perturbed by Lévy Noise
The large deviations theory is among the most classical areas in probability theory with many deep developments and applications. Several authors have established the Wentzell-Freidlin type large deviation estimates for a class of infinite dimensional stochastic differential equations (see for eg., Budhiraja and Dupuis [9], Da Prato and Zabczyk [12], Kallianpur and Xiong [20]). In these works t...
متن کاملExponential Stability in Mean Square of Neutral Stochastic Differential Difference Equations
In this paper we shall discuss the exponential stability in mean square for a neutral stochastic diierential diierence equation of the form dx(t) ? G(x(t ?))] = f(t; x(t); x(t ?))dt + (t; x(t); x(t ?))dw(t). In the case when (t; x; y) 0 this neutral stochastic diierential diierence equations becomes a deterministic neutral diierential diierence equations d dt x(t)?G(x(t?))] = f(t; x(t); x(t?))....
متن کاملRare Events in Stochastic Partial Differential Equations on Large Spatial Domains
A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptoti...
متن کاملAlmost Sure Exponential Stability of Neutral Stochastic Diierential Diierence Equations 1
A neutral stochastic diierential diierence equation
متن کاملA PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS
A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...
متن کامل