scholarly journals An Elementary Derivation of the Large Deviation Rate Function for Finite State Markov Chains

2013 ◽  
Vol 16 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Mathukumalli Vidyasagar
Keyword(s):  
Markov Chains ◽  
Large Deviation ◽  
Rate Function ◽  
Elementary Derivation ◽  
Deviation Rate Function ◽  
Deviation Rate ◽  
Finite State

scholarly journals ANALYTICAL MECHANICS IN STOCHASTIC DYNAMICS: MOST PROBABLE PATH, LARGE-DEVIATION RATE FUNCTION AND HAMILTON–JACOBI EQUATION

2012 ◽  
Vol 26 (24) ◽  
pp. 1230012 ◽  
Author(s):  
HAO GE ◽  
HONG QIAN
Keyword(s):  
Jacobi Equation ◽  
Stochastic Dynamics ◽  
Large Deviation ◽  
Applied Mathematics ◽  
Rate Function ◽  
Analytical Mechanics ◽  
Stochastic Motion ◽  
Deviation Rate Function ◽  
Deviation Rate ◽  
Hamilton Jacobi Equation

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Lagrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation dXt = b(Xt)dt+ϵdWt, where Wt is a Brownian motion. In the limit of vanishingly small ϵ, the solution to the stochastic differential equation other than [Formula: see text] are all rare events. However, conditioned on an occurrence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with [Formula: see text] and Hamiltonian equations with H(p, q) = ‖p‖2+b(q)⋅p. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for Xt as f(x, t) = e-u(x, t)/ϵ, where u(x, t) is called a large-deviation rate function which satisfies the corresponding Hamilton–Jacobi equation. An irreversible diffusion process with ∇×b≠0 corresponds to a Newtonian system with a Lorentz force [Formula: see text]. The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions and integrable systems.

scholarly journals Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains

2018 ◽  
Author(s):  
Rodrigo Cofré ◽  
Cesar Maldonado ◽  
Fernando Rosas
Keyword(s):  
Markov Chains ◽  
Large Deviations ◽  
Maximum Entropy ◽  
Large Deviation ◽  
Spike Trains ◽  
Rate Function ◽  
Statistical Fluctuation ◽  
Maximum Entropy Models ◽  
Deviation Rate Function ◽  
Neuronal Spike Trains

We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. We review large deviations techniques useful in this context to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.

scholarly journals STEADY STATE FLUCTUATIONS OF THE DISSIPATED HEAT FOR A QUANTUM STOCHASTIC MODEL

2006 ◽  
Vol 18 (06) ◽  
pp. 619-653 ◽  
Author(s):  
WOJCIECH DE ROECK ◽  
CHRISTIAN MAES
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Stochastic Dynamics ◽  
Large Deviation ◽  
Rate Function ◽  
Entropy Dissipation ◽  
Deviation Rate Function ◽  
Deviation Rate ◽  
Different Temperatures ◽  
Multi Level

We introduce a quantum stochastic dynamics for heat conduction. A multi-level subsystem is coupled to reservoirs at different temperatures. Energy quanta are detected in the reservoirs allowing the study of steady state fluctuations of the entropy dissipation. Our main result states a symmetry in its large deviation rate function.

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