scholarly journals THE MATHER MEASURE AND A LARGE DEVIATION PRINCIPLE FOR THE ENTROPY PENALIZED METHOD

2011 ◽  
Vol 13 (02) ◽  
pp. 235-268 ◽  
Author(s):  
D. A. GOMES ◽  
A. O. LOPES ◽  
J. MOHR
Keyword(s):  
Jacobi Equation ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Deviation Function ◽  
Deviation Principle ◽  
Probability Densities ◽  
Aubry Set ◽  
Dynamical Properties ◽  
Hamilton Jacobi Equation

We present the rate function and a large deviation principle for the entropy penalized Mather problem when the Lagrangian is generic (it is known that in this case the Mather measure μ is unique and the support of μ is the Aubry set). We assume the Lagrangian L(x, v), with x in the torus 𝕋N and v∈ℝN, satisfies certain natural hypotheses, such as superlinearity and convexity in v, as well as some technical estimates. Consider, for each value of ϵ and h, the entropy penalized Mather problem [Formula: see text] where the entropy S is given by [Formula: see text], and the minimization is performed over the space of probability densities μ(x, v) on 𝕋N×ℝN that satisfy the discrete holonomy constraint ∫𝕋N×ℝN φ(x + hv) - φ(x) dμ = 0. It is known [17] that there exists a unique minimizing measure μϵ, h which converges to a Mather measure μ, as ϵ, h→0. In the case in which the Mather measure μ is unique we prove a Large Deviation Principle for the limit lim ϵ, h→0ϵ ln μϵ, h(A), where A ⊂ 𝕋N×ℝN. In particular, we prove that the deviation function I can be written as [Formula: see text], where ϕ0 is the unique viscosity solution of the Hamilton – Jacobi equation, [Formula: see text]. We also prove a large deviation principle for the limit ϵ→ 0 with fixed h. Finally, in the last section, we study some dynamical properties of the discrete time Aubry–Mather problem, and present a proof of the existence of a separating subaction.

scholarly journals The Large Deviations of Estimating Rate Functions

2005 ◽  
Vol 42 (01) ◽  
pp. 267-274 ◽  
Author(s):  
Ken Duffy ◽  
Anthony P. Metcalfe
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Length Distribution ◽  
Rate Function ◽  
Partial Sums ◽  
Single Server ◽  
Mixing Condition ◽  
Deviation Principle ◽  
Queue Length Distribution ◽  
Empirical Estimates

Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

scholarly journals The Large Deviation Principle for the On-Off Weibull Sojourn Process

2008 ◽  
Vol 45 (01) ◽  
pp. 107-117 ◽  
Author(s):  
Ken R. Duffy ◽  
Artem Sapozhnikov
Keyword(s):  
Renewal Process ◽  
Time Scale ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Compact Set ◽  
Natural Processes ◽  
Deviation Principle ◽  
Rate Functions ◽  
Nonlinear Scale

This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.

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 The Large Deviation Principle for the On-Off Weibull Sojourn Process

2008 ◽  
Vol 45 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Ken R. Duffy ◽  
Artem Sapozhnikov
Keyword(s):  
Renewal Process ◽  
Time Scale ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Compact Set ◽  
Natural Processes ◽  
Deviation Principle ◽  
Rate Functions ◽  
Nonlinear Scale

This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.

scholarly journals The Large Deviations of Estimating Rate Functions

2005 ◽  
Vol 42 (1) ◽  
pp. 267-274 ◽  
Author(s):  
Ken Duffy ◽  
Anthony P. Metcalfe
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Length Distribution ◽  
Rate Function ◽  
Partial Sums ◽  
Single Server ◽  
Mixing Condition ◽  
Deviation Principle ◽  
Queue Length Distribution ◽  
Empirical Estimates

Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

THE LARGE DEVIATION PRINCIPLE FOR MEASURES WITH RANDOM WEIGHTS

1993 ◽  
Vol 05 (04) ◽  
pp. 659-692 ◽  
Author(s):  
R. S. ELLIS ◽  
J. GOUGH ◽  
J. V. PULÉ
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Spherical Model ◽  
Fermi Gas ◽  
Rate Function ◽  
Bose Statistics ◽  
Deviation Principle ◽  
Random Weights ◽  
Abstract Setting ◽  
Special Case

In this paper, we study the problem of large deviations for measures with random weights. We are motivated by previous work dealing with the special case occuring in the statistical mechanics of the Bose gas. We study the problem in an abstract setting, isolating what is general from what is dependent on Bose statistics. We succeed in proving the large deviation principle for a large class of measures with random weights and obtaining the corresponding rate function in an explicit form. In particular, our results are applicable to the Fermi gas and the spherical model.

An invariance principle and a large deviation principle for the biased random walk on

2020 ◽  
Vol 57 (1) ◽  
pp. 295-313
Author(s):  
Yuelin Liu ◽  
Vladas Sidoravicius ◽  
Longmin Wang ◽  
Kainan Xiang
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Invariance Principle ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Scaling Limit ◽  
Rate Function ◽  
Deviation Principle ◽  
Biased Random Walk

AbstractWe establish an invariance principle and a large deviation principle for a biased random walk ${\text{RW}}_\lambda$ with $\lambda\in [0,1)$ on $\mathbb{Z}^d$ . The scaling limit in the invariance principle is not a d-dimensional Brownian motion. For the large deviation principle, its rate function is different from that of a drifted random walk, as may be expected, though the reflected biased random walk evolves like the drifted random walk in the interior of the first quadrant and almost surely visits coordinate planes finitely many times.

scholarly journals Large deviations for the Ornstein-Uhlenbeck process with shift

2015 ◽  
Vol 47 (03) ◽  
pp. 880-901 ◽  
Author(s):  
Bernard Bercu ◽  
Adrien Richou
Keyword(s):  
Maximum Likelihood ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Maximum Likelihood Estimates ◽  
Shift Parameter ◽  
Uhlenbeck Process ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Ornstein Uhlenbeck Process

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

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