Henry P. McKean

The central limit theorem for Carleman's equation

The purpose of this paper is to discuss the analogy between the law of large numbers and the central limit theorem of classical probability theory on the one hand and the hydrodynamical approximations in the statistical mechanics of gases on the other. The chief illustration is provided by Carlemans model [2] for which the central limit approximation is a kind of non-linear Brownian motion regulated by ∂n/∂t=(n/n)′.

Geometry of KdV (1): Addition and the Unimodular Spectral Classes

This is the first of three papers on the geometry of KDV. It presents what purports to be a foliation of an extensive function space into which all known invariant manifolds of KDV fit naturally as special leaves. The two main themes are addition (each leaf has its private one) and unimodular spectral classes (each leaf has a spectral interpretation), but first a bit of background.

A Martin boundary connected with the ∞-volume limit of the focussing cubic Schrödinger equation

The existence of a change of phase in the micro-canonical ensemble for the focussing cubic Schrodinger system was suggested by Lebowitz-Rose-Speer [1989]. Chorin [1994] disputes their numerical evidence; his own is based on a more sophisticated approximation to the micro-canonical distribution and leads him to the opposite conclusion. Perhaps the source of this contradictory testimony is the fact, proved here, that the ∞-volume limit does not exist at any temperature 0 < T < ∞ or density 0 < D < ∞. This does not preclude a more or less dramatic change in the ensemble, from high to low temperatures, but it does guarantee the existence of several distinct ∞-volume Gibbs states. These are related to a sort of “boundary” of the type introduced by Martin [1941] for the description of classical harmonic functions in general 3-dimensional regions, as will be explained below.

Local and inverse local times

We now take up the fine structure of the local time t(t) = t(t 0) and its inverse function t-1(t) for a persistent non-singular diffusion D* on an interval Q containing 0 as an inside point or as a left end point, with — u+(0) + m(0) (Gu)(0) = 0 in the second case. A number of the statements made below hold for transient diffusions also (see esp. 6.3, 6.5, 6.6); the necessary modifications of the proofs are left to the reader.

The general 1-dimensional diffusion

Roughly speaking, a 1-dimensional diffusion is a model of the (stochastic) motion of a particle with life time m ∞ ≦ + ∞ (also called killing time), continuous path x(t): t< m ∞ and no memory, travelling in a linear interval Q; the phrase no memory is meant to suggest that the motion starts afresh at certain (Markov) times including all the constant times m ≡ s ≧ 0, i.e., if m is a Markov time, then, conditional on the present position x(m), the statistical properties of the future motion x (t + m) (t≧ 0) do not depend upon the past x (t) (t ≦ m).

A general view of diffusion in several dimensions

Given a (conservative) diffusion D on a space Q as described in 7.1, its generator G can be expressed in terms of the hitting probabilities and mean exit times n n1a) n nh(a,db)= h əD (a,db) = Pa[x(MəD)∈ db, MəD <+∞] n n n n n1b) n ne(a) = e D ) (a) = E a ( MəD) n n n n n n nfor open D⊂Q via E. B. Dynkin’s formula n n

Potentials and the random walk

= mathop {lim }limits_{D downarrow a} e{left( a right)^{ - 1}}left[ {mathop smallint limits_{partial D} hleft( {a,db} right)uleft( b right) - uleft( a right)} right]

The central limit theorem for Carlemans equation

n n(2) n nto borrow a phrase of W. Feller’s, h is the road map, i.e., it tells what routes the particle is permitted to travel, and e governs the speed.