scholarly journals On the Large Deviation Rate Function for the Empirical Measures of Reversible Jump Markov Processes

2013 ◽  
Author(s):  
Paul Dupuis ◽  
Yufei Liu
Keyword(s):  
Markov Processes ◽  
Large Deviation ◽  
Rate Function ◽  
Reversible Jump ◽  
Empirical Measures ◽  
Deviation Rate Function ◽  
Deviation Rate ◽  
Jump Markov Processes

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 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.

scholarly journals The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities

2020 ◽  
Vol 178 (1-2) ◽  
pp. 475-530 ◽  
Author(s):  
Julio Backhoff ◽  
Giovanni Conforti ◽  
Ivan Gentil ◽  
Christian Léonard
Keyword(s):  
Large Deviation ◽  
Mean Field ◽  
Rate Function ◽  
Convergence To Equilibrium ◽  
Functional Inequalities ◽  
Brownian Particles ◽  
Interesting Connection ◽  
Deviation Rate Function ◽  
Ergodic Behavior ◽  
The Mean

Abstract We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian particles conditionally on the observation of their initial and final configuration. Its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent Brownian particles. We undertake a fine study of the dynamics of its solutions, including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger, as well as a novel class of functional inequalities involving the mean field entropic cost (i.e. the optimal value in (MFSP)). Our strategy unveils an interesting connection between forward backward stochastic differential equations and the Riemannian calculus on the space of probability measures introduced by Otto, which is of independent interest.

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