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Large Deviation Theory
Published in Teunis C. Dorlas, Statistical Mechanics, 2021
We want to generalize the theory of the previous chapter to particles with more than two energy levels. To this end we shall first generalize Laplace’s theorem to a result more suited to the general situation. This generalization is called large-deviation theory. In chapter 21 we apply this theory in re-deriving equation (19.10) for the entropy of a paramagnet. In chapter 22 we consider the more general situation. Large-deviation theory also has important applications in other areas of mathematics, for example information theory, dynamical systems, and queueing theory. See problems II-19 to II-23.
Introduction
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Erchin Serpedin, Thomas Chen, Dinesh Rajan
Chapter 9 is dedicated to the large deviation theory, i.e., the study of occurrence of rare events. This chapter represents a short introduction to the basic large deviation concepts and techniques including concentration inequalities, rate function, Cramer’s theorem, type analysis, and Sanov’s theorem. Determination of the error exponents of several standard hypothesis testing methods is presented as an application of large deviation theory.
Large deviation principles for a 2D liquid crystal model with jump noise
Published in Applicable Analysis, 2022
The purpose of this is to establish a large deviation principle (LDP) for a 2D stochastic EL crystal flow driven by a jump noise of Lévy type. As we know, the large deviation theory is concerned with the study of the precise asymptotic behavior governing the decay rate of probabilities of rare events. The LDP for partial differential equations such as the Navier–Stokes equations are investigated in several articles, [23, 24]. A LDP for SPDE with Gaussian noise have been investigated in several articles such [25, 26]. In these papers, the LDP is usually obtained using the weak convergence approach introduced in [27–29]. There is not that much study on the LDP for SPDE with multiplication noise of Lévy type. In [30–32], the authors studied the LDP for SPDE with models such as the Navier–Stokes equations with multiplication noise of Lévy type. The presence of the jump in these models makes the analysis more involved. In these papers, the proof of LDP is usually based on the weak convergence method introduced in [30]. In [33], the authors established a LDP for a 2D simplified stochastic EL model with a Ginzburg-Landau approximation. One main difference of our work and that of [33] is that in the model considered in [33] the constraint is removed and the term is replaced by a polynomial that satisfies some reasonable growth assumptions. Another important difference is that our model includes a more general noise, both in the equations for the velocity v as well as the director field We note that in [33], the noise term appears only in the momentum equation.