Remarks on the vanishing discount problem for infinite systems of Hamilton–Jacobi–Bellman equations

This paper is concerned with the asymptotic analysis of infinite systems of weakly coupled stationary Hamilton–Jacobi–Bellman equations as the discount factor tends to zero. With a specific Hamiltonian, we show the convergence of the solution and prove the solvability of the corresponding ergodic problem.

Ergodic pairs for singular or degenerate fully nonlinear operators

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub-solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.

Singular perturbations for a subelliptic operator

We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.

A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

ON THE FOUNDATIONS OF STATISTICAL MECHANICS: A BRIEF REVIEW

This work is dedicated to those who are genuinely interested in the foundations of statistical mechanics. Statistical mechanics is a very successful theory. The foundations on which it is based are radically different from the foundations of Newtonian mechanics. Probability and the law of large numbers are its fundamental ingredients. In this paper we will see how the statistical mechanical synthesis which incorporates mechanics (deterministic) and probability theory (indeterministic) works. The approach consists of replacing the actual system by a hypothetical ensemble of systems under same external conditions. From the most used prescription of Gibbs, we calculate the phase space averages of dynamical quantities and find that these phase averages agree very well with experiments. Clearly, actual experiments are not done on a hypothetical ensemble. They are done on the actual system in the laboratory and these experiments take a finite amount of time. Thus, it is usually argued that actual measurements are time averages and they are equal to phase averages due to ergodic hypothesis (time averages = phase space averages). The aim of the present review is to show that ergodic hypothesis is not relevant for equilibrium statistical mechanics (with Tolman and Landau). We will see that the solution for the problem is in the very peculiar nature of the macroscopic observables and with a very large number of degrees of freedom involved in macroscopic systems as first pointed out by Boltzmann and then more quantitatively by Khinchin. Similar arguments were used by Landau based upon the approximate property of "statistical independence". We analyze these ideas in detail. We present a critique of the ideas of Jaynes who says that the ergodic problem is a conceptual one and is related to the very concept of ensemble itself which is a by-product of the frequency theory of probability, and the ergodic problem becomes irrelevant when the probabilities of various microstates are interpreted with Laplace–Bernoulli Theory of Probability (Bayesian viewpoint). At the end, we critically review various quantum approaches (quantum-statistical typicality approaches) to the foundations of statistical mechanics. The literature on quantum-statistical typicality is organized under four notions (i) kinematical canonical typicality, (ii) dynamical canonical typicality, (iii) kinematical normal typicality, and (iv) dynamical normal typicality. Analogies are seen in the Khinchin's classical approach and in the modern quantum-statistical typicality approaches.