Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Topics in propagation of chaos

The large deviation approach to statistical mechanics

We consider stochastic dynamic games in large population conditions where multiclass agents are weakly coupled via their individual dynamics and costs. We approach this large population game problem by the so-called Nash Certainty Equivalence (NCE) Principle which leads to a decentralized control synthesis. The McKean-Vlasov NCE method presented in this paper has a close connection with the statistical physics of large particle systems: both identify a consistency relationship between the individual agent (or particle) at the microscopic level and the mass of individuals (or particles) at the macroscopic level. The overall game is decomposed into (i) an optimal control problem whose Hamilton-Jacobi-Bellman (HJB) equation determines the optimal control for each individual and which involves a measure corresponding to the mass effect, and (ii) a family of McKean-Vlasov (M-V) equations which also depend upon this measure. We designate the NCE Principle as the property that the resulting scheme is consistent (or soluble), i.e. the prescribed control laws produce sample paths which produce the mass effect measure. By construction, the overall closed-loop behaviour is such that each agent’s behaviour is optimal with respect to all other agents in the game theoretic Nash sense.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cis/2006/0006/0003/CIS-2006-0006-0003-a005.pdf

Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

A system of N diffusions on R$SUP:D$ESUP:in which the interaction is expressed in terms of the empirical measure is considered. The limiting behavior as N →∞ is described by a McKean_Vlasov equation. The purpose of this paper is to show that the large deviations from the McKean-Vlasov limit can be described by a generalization of the theory of Freidlin and Wentzell and to obtain a characterization of the action functional. In order to obtain this action functional we first obtain results on projective limits of large deviation systems, large deviations on dual vector spaces and a Sanov type theorem for vectors of empirical measures

Large deviations from the mckean-vlasov limit for weakly interacting diffusions

On Large Deviations from the Invariant Measure

For a system of diffusions in a domain of Rd with long-range weak interaction the behavior of the associated empirical process is studied. Under mild growth and smoothness assumptions on the drift and diffusion coefficients such as coercivity and monotonicity conditions the law of large numbers and the propagation of chaos are proved. Existence and uniqueness of the weak solution to the McKean - Vlasov equation and the associated non-linear martingale problem are investigated.

On the McKean-Vlasov Limit for Interacting Diffusions

The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a qualitative level, intermittent random fields are distinguished by the appearance of sparsely distributed sharp “peaks” which give the main contribution to the formation of the statistical moments. The paper deals with the Cauchy problem (∂/∂t)u(t,x)=Hu(t, x), u(0,x)=t
0(x) ≥ 0, (t, x) ∈ ℝ+ × ℤ
 d
 , for the Anderson HamiltonianH = κΔ + ξ(·), ξ(x),x ∈ ℤd where is a (generally unbounded) spatially homogeneous random potential. This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fieldsu(t,·) ast→∞ are found in spectral terms ofH. Rough asymptotic formulas for the statistical moments and the almost sure behavior ofu(t,x) ast→∞ are also derived.

/pdf/parabolic-problems-for-the-anderson-model-i-intermittency-yd7wl5eyno.pdf

Parabolic problems for the Anderson model. I. Intermittency and related topics

This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials We thereby consider examples of potentials studied in the literature In the particularly important case of an iid potential with double-exponential tails we formulate the asymptotic results in detail Furthermore, we explain that, under mild regularity assumptions, there are only four different universality classes of asymptotic behaviors Finally, we study the moment Lyapunov exponents for space-time homogeneous catalytic potentials generated by a Poisson field of random walks

Jürgen Gärtner

Papers

Large deviations from the mckean-vlasov limit for weakly interacting diffusions

On Large Deviations from the Invariant Measure

On the McKean-Vlasov Limit for Interacting Diffusions

Parabolic problems for the Anderson model. I. Intermittency and related topics

The Parabolic Anderson Model