Hammersley's interacting particle process and longest increasing subsequences

Summary In a famous paper [8] Hammersley investigated the lengthL _n of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly “soft” arguments that limn ^′1/2 EL _n =2. This is a known result, but previous proofs [14, 11] relied on hard analysis of combinatorial asymptotics. Research supported by NSF Grant MCS 92-24857 and the Miller Institute for Basic Research in Science Research supported by NSF Grant DMS92-04864     

One-point extensions of Markov processes by darning

This paper is a continuation of the works by Fukushima–Tanaka (Ann Inst Henri Poincaré Probab Stat 41: 419–459, 2005) and Chen–Fukushima–Ying (Stochastic Analysis and Application, p.153–196. The Abel Symposium, Springer, Heidelberg) on the study of one-point extendability of a pair of standard Markov processes in weak duality. In this paper, general conditions to ensure such an extension are given. In the symmetric case, characterizations of the one-point extensions are given in terms of their Dirichlet forms and in terms of their L ²-infinitesimal generators. In particular, a generalized notion of flux is introduced and is used to characterize functions in the domain of the L ²-infinitesimal generator of the extended process. An important role in our investigation is played by the α-order approaching probability u _α . The research of Z.-Q. Chen is supported in part by NSF Grant DMS-0600206. The research of M. Fukushima is supported in part by Grant-in-Aid for Scientific Research of MEXT No.19540125.     

Loop-erased random walk on the Sierpinski gasket

In this paper the loop-erased random walk on the finite pre-Sierpiński gasket is studied. It is proved that the scaling limit exists and is a continuous process. It is also shown that the path of the limiting process is almost surely self-avoiding, while having Hausdorff dimension strictly greater than 1. The loop-erasing procedure proposed in this paper is formulated by erasing loops, in a sense, in descending order of size. It enables us to obtain exact recursion relations, making direct use of ‘self-similarity’ of a fractal structure, instead of the relation to the uniform spanning tree. This procedure is proved to be equivalent to the standard procedure of chronological loop-erasure.     

Sets versus trial sequences,Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920

The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix) of two critical letters of November 1919 by the “pure” mathematician Felix Hausdorff to the engineer and applied mathematician Richard von Mises compose about one third of the paper. The article also investigates von Mises’s ill-conceived effort to adopt measures and his misinterpretation of an influential book of Constantin Carathéodory. A short and sketchy look at the subsequent development of the standpoints of the pure and the applied mathematician—here represented by Hausdorff and von Mises—in the probability theory of the 1920s and 1930s concludes the paper.     

Convergence rates in the strong laws of asymptotically negatively associated random fields

In this paper, a notion of negative side p-mixing (p -mixing) which can be regardedas asymptotic negative association is defined, and some Rosenthal type inequalities for p -mix-ing random fields are established. The complete convergence and almost sure summability onthe convergence rates with respect to the strong law of large numbers are also discussed for p--mixing random fields. The results obtained extend those for negatively associated sequences andp“ -mixing random fields.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

Large deviations and related problems for absorbing Markov chains

In this paper, large deviations and their connections with several other fundamental topics are investigated for absorbing Markov chains. A variational representation for the Dirichlet principal eigenvalues is given by the large deviation approach. Kingman’s decay parameters and mean ratio quasi-stationary distributions of the chains are also characterized by the large deviation rate function. As an application of these results, we interpret the “stationarity” of mean ratio quasi-stationary distributions via a concrete example. An application to quasi-ergodicity is also discussed.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the uniform norm of solutions of quasilinear SPDE's

In this paper we prove L^p estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.     

Itô’s stochastic calculus: Its surprising power for applications

We trace Itô’s early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Itô’s formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Itô’s formula in mathematical finance in the 1970s. Throughout the paper, we treat Itô’s jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions.     

A mixed-step algorithm for the approximation of the stationary regime of a diffusion

In some recent papers, some procedures based on some weighted empirical measures related to decreasing-step Euler schemes have been investigated to approximate the stationary regime of a diffusion (possibly with jumps) for a class of functionals of the process. This method is efficient but needs the computation of the function at each step. To reduce the complexity of the procedure (especially for functionals), we propose in this paper to study a new scheme, called the mixed-step scheme, where we only keep some regularly time-spaced values of the Euler scheme. Our main result is that, when the coefficients of the diffusion are smooth enough, this alternative does not change the order of the rate of convergence of the procedure. We also investigate a Richardson–Romberg method to speed up the convergence and show that the variance of the original algorithm can be preserved under a uniqueness assumption for the invariant distribution of the “duplicated” diffusion, condition which is extensively discussed in the paper. Finally, we conclude by giving sufficient “asymptotic confluence” conditions for the existence of a smooth solution to a discrete version of the associated Poisson equation, condition which is required to ensure the rate of convergence results.     

The Itô–Nisio theorem,quadratic Wiener functionals,and 1-solitons

Among Professor Kiyosi Itô’s achievements, there is the Itô–Nisio theorem, a completely general theorem relative to the Fourier series decomposition of Brownian motion. In this paper, some of its applications will be reviewed, and new applications to 1-soliton solutions to the Korteweg–de Vries (KdV for short) equation and Eulerian polynomials will be given.     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

A criterion for a continuous spectral density

For a given weakly stationary random field indexed by the integer lattice of an arbitrary finite dimension, a necessary and sufficient condition is given for the existence of a continuous spectral density. The condition involves the covariances of pairs of sums of the random variables, with the two index sets being “separated” from each other (but possibly “interlaced”) by a certain distance along a coordinate direction.     

Properties of solutions of stochastic differential equations with continuous-state-dependent switching

This work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions. The word “hybrid” highlights the coexistence of continuous dynamics and discrete events. The underlying process has two components. One component describes the continuous dynamics, whereas the other is a switching process representing discrete events. One of the main features is the switching component depending on the continuous dynamics. In this paper, weak continuity is proved first. Then continuous and smooth dependence on initial data are demonstrated. In addition, it is shown that certain functions of the solutions verify a system of Kolmogorov's backward differential equations. Moreover, rates of convergence of numerical approximation algorithms are dealt with.     

N/V-limit for stochastic dynamics in continuous particle systems

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ ^d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ ^d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.     

The boundary value problem for discrete analytic functions

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.     

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤ^ν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in [BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2. Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001     

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

On the conditioned exit measures of super Brownian motion

In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general, as the change of measure is given by a martingale which need not arise from a single harmonic function. Received: 27 August 1998 / Revised version: 8 January 1999     

Skorokhod embeddings, minimality and non-centred target distributions

In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality. We then apply these results, firstly to the problem of embedding non-centred target distributions in Brownian motion, and secondly to embedding general target laws in a diffusion. We construct an embedding (which reduces to the Azema-Yor embedding in the zero-target mean case) which maximises the law of sup_s≤TB_s among the class of minimal embeddings of a general target distribution μ in Brownian motion. We then construct a minimal embedding of μ in a diffusion X which maximises the law of sup_s≤Th(X_s) for a general function h.