Ray Knight Research Articles

Ray–Knight compactification of birth and death processes

A birth and death process is a continuous-time Markov chain with minimal state space N, whose transition matrix is standard and whose density matrix is a birth–death matrix. Birth and death process is unique if and only if ∞ is an entrance or natural. When ∞ is neither an entrance nor natural, there are two ways in the literature to obtain all birth and death processes. The first one is an analytic treatment proposed by Feller in 1959, and the second one is a probabilistic construction completed by Wang in 1958.In this paper we will give another way to study birth and death processes using the Ray–Knight compactification. This way has the advantage of both the analytic and probabilistic treatments above. By applying the Ray–Knight compactification, every birth and death process can be modified into a càdlàg Ray process on N∪{∞}∪{∂}, which is either a Doob processes or a Feller Q-process. Every birth and death process in the second class has a modification that is a Feller process on N∪{∞}∪{∂}. We will derive the expression of its infinitesimal generator, which explains its boundary behaviours at ∞. Furthermore, by using the killing transform and the Ikeda–Nagasawa–Watanabe piecing out procedure, we will also provide a probabilistic construction for birth and death processes. This construction relies on a triple determining the resolvent matrix introduced by Wang and Yang in their work (Wang and Yang, 1992).

A Ray–Knight theorem for ∇ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla \\phi $$\\end{document} interface models and scaling limits

We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which—generically—is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray–Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R^3$$\\end{document} with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.

Ray Knight theorems for spectrally negative Lévy processes

Ray Knight theorems for spectrally negative Lévy processes

Hidden symmetries and limit laws in the extreme order statistics of the Laplace random walk

This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett. 108 (2012) 040601). In particular, the Bessel process of dimension 4=2+2 appears in the descriptions as a path decomposition of Brownian motion at a local minimum and the Ray–Knight description of Brownian local times near the minimum.

A Ray–Knight representation of up-down Chinese restaurants

We study composition-valued continuous-time Markov chains that appear naturally in the framework of Chinese Restaurant Processes (CRPs). As time evolves, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (Ann. Probab. 37 (2009) 1999–2041). We relate such up-down CRPs to the splitting trees of Lambert (Ann. Probab. 38 (2010) 348–395) inducing spectrally positive Lévy processes. Conversely, we develop theorems of Ray–Knight type to recover more general up-down CRPs from the heights of Lévy processes with jumps marked by integer-valued paths. We further establish limit theorems for the Lévy process and the integer-valued paths to connect to work by Forman, Pal, Rizzolo, Shi and Winkel on interval partition diffusions and hence to some long-standing conjectures.

Convergence of random walks with Markovian cookie stacks to Brownian motion perturbed at extrema

We consider one-dimensional excited random walks (ERWs) with i.i.d. Markovian cookie stacks in the non-boundary recurrent regime. We prove that under diffusive scaling such an ERW converges in the standard Skorokhod topology to a multiple of Brownian motion perturbed at its extrema (BMPE). All parameters of the limiting process are given explicitly in terms of those of the cookie Markov chain at a single site. While our results extend the results in Dolgopyat and Kosygina (Electron Commun Probab 17:1–14, 2012) (ERWs with boundedly many cookies per stack) and Kosygina and Peterson (Electron J Probab 21:1–24, 2016) (ERWs with periodic cookie stacks), the approach taken is very different and involves coarse graining of both the ERW and the random environment changed by the walk. Through a careful analysis of the environment left by the walk after each “mesoscopic” step, we are able to construct a coupling of the ERW at this “mesoscopic” scale with a suitable discretization of the limiting BMPE. The analysis is based on generalized Ray–Knight theorems for the directed edge local times of the ERW stopped at certain stopping times and evolving in both the original random cookie environment and (which is much more challenging) in the environment created by the walk after each “mesoscopic” step.

The Height Process of a Continuous-State Branching Process with Interaction

For a generalized continuous-state branching process with non-vanishing diffusion part, finite expectation and a directed (“left-to-right”) interaction, we construct the height process of its forest of genealogical trees. The connection between this height process and the population size process is given by an extension of the second Ray–Knight theorem. This paper generalizes earlier work of the two last authors which was restricted to the case of continuous branching mechanisms. Our approach is different from that of Berestycki et al. (Probab Theory Relat Fields 172:725–788, 2018). There the diffusion part of the population process was allowed to vanish, but the class of interactions was more restricted.

Mixture representations of noncentral distributions

With any symmetric distribution μ on the real line we may associate a parametric family of noncentral distributions as the distributions of where X is a random variable with distribution μ. The classical case arises if μ is the standard normal distribution, leading to the noncentral chi-squared distributions. It is well known that these may be written as Poisson mixtures of the central chi-squared distributions with odd degrees of freedom. We obtain such mixture representations for the logistic distribution and for the hyperbolic secant distribution. We also derive alternative representations for chi-squared distributions and relate these to representations of the Poisson family. While such questions originated in parametric statistics they also appear in the context of the generalized second Ray–Knight theorem, which connects Gaussian processes and local times of Markov processes.

On the local time process of a skew Brownian motion

We derive a Ray–Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both of these cases. The Ray–Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper.

On pinned fields, interlacements, and random walk on $$({\mathbb {Z}}/N {\mathbb {Z}})^2$$ ( Z / N Z ) 2

We define two families of Poissonian soups of bidirectional trajectories on $${\mathbb {Z}}^2$$ , which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus $$({\mathbb {Z}}/N {\mathbb {Z}})^2$$ , started from the uniform distribution, run up to a time of order $$(N\log N)^2$$ and forced to avoid a fixed point. The local limit of the latter was recently established in Comets et al. (Commun Math Phys 343:129–164, 2016). Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of “finite volume” approximations, consisting of random walks avoiding the origin and killed at spatial scale N, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning the intensity u of such walks with N, the occupation field can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Our construction thus yields a two-dimensional analogue of the random interlacements model introduced in Sznitman (Ann Math 171(3):2039–2087, 2010) in the transient case. It also links it to the pinned free field in $${\mathbb {Z}}^2$$ , by means of a (pinned) Ray–Knight type isomorphism theorem.

Squared Bessel processes of positive and negative dimension embedded in Brownian local times

The Ray–Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \mu \ell $ of a reflecting Brownian motion $|B|$ by a multiple $\mu $ of its local time process $\ell $ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B$. This is done by decomposing the path of $B$ into its excursions above and below a family of continuous random levels determined by the Harrison–Shepp construction of skew Brownian motion as the strong solution of an SDE driven by $B$. This embedding connects to Brownian local times a framework of point processes of squared Bessel excursions of negative dimension and associated stable processes, recently introduced by Forman, Pal, Rizzolo and Winkel to set up interval partition evolutions that arise in their approach to the Aldous diffusion on a space of continuum trees.

Ray–Knight representation of flows of branching processes with competition by pruning of Lévy trees

We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert (Ann Appl Probab 15(2):1506–1535, 2005). Following the approach developed by Dawson and Li (Ann Probab 40(2):813–857, 2012), we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Levy-trees, and establish a Ray–Knight representation result for these processes in terms of the local times of suitably pruned forests.

Addicted. pregnant. poor. By Kelly Ray Knight

Book review.

A limit theorem for moments in space of the increments of Brownian local time

We prove a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. [Ann. Prob. 38 (2010) 396–438] and Rosen [Stoch. Dyn. 11 (2011) 5–48], which were later reproven by Hu and Nualart [Electron. Commun. Probab. 15 (2010) 396–410] and Rosen [In Séminaire de Probabilités XLIII (2011) 95–104 Springer] are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary orders. The main ingredients are Perkins’ semimartingale decomposition, the Kailath–Segall identity and an asymptotic Ray–Knight theorem by Pitman and Yor.

Review of <em>addicted. pregnant. poor.</em> Kelly Ray Knight. Reviewed by Cara Kelly

Review of <em>addicted. pregnant. poor.</em> Kelly Ray Knight. Reviewed by Cara Kelly

Branching processes with interaction and a generalized Ray–Knight Theorem

Nous considerons un modele d’evolution d’une population avec interaction entre les individus, ou les taux de naissance et de mort sont fonction de la taille de la population. Nous obtenons la limite en grande population apres renormalisation, qui est solution de l’EDS \[Z^{x}_{t}=x+\int_{0}^{t}f(Z^{x}_{s})\,\mathrm{d}s+2\int_{0}^{t}\int_{0}^{Z^{x}_{s}}W(\mathrm{d}s,\mathrm{d}u),\] ou $W(\mathrm{d}s,\mathrm{d}u)$ est un bruit blanc sur $[0,\infty)^{2}$. Nous donnons une representation de cette diffusion a la Ray–Knight, en fonction des temps locaux d’un mouvement brownien reflechi $H$ avec une derive qui depend du temps local accumule par $H$ a son niveau courant, a travers la fonction $f'/2$.

Sub-Gaussian bound for the one-dimensional Bouchaud trap model

We establish a sub-Gaussian lower bound for the transition kernel of the one-dimensional, symmetric Bouchaud trap model, which provides a positive answer to the behavior predicted by Bertin and Bouchaud in (Phys. Rev. E (3) 67 (2013) 026128). The proof rests on the Ray–Knight description of the local time of a one-dimensional Brownian motion. Using the same ideas, we also prove the corresponding result for the FIN singular diffusion.

A new proof of an Engelbert–Schmidt type zero–one law for time-homogeneous diffusions

Abstract In this paper we give a new proof to an Engelbert–Schmidt type zero–one law for time-homogeneous diffusions, which provides deterministic criteria for the convergence of integral functional of diffusions. Our proof is based on a slightly stronger assumption than that in Mijatovic and Urusov (2012a) and utilizes stochastic time change and Feller’s test of explosions. It does not rely on advanced methods such as the first Ray–Knight theorem, William’s theorem, Shepp’s dichotomy result for Gaussian processes or Jeulin’s lemma as in the previous literature (see Mijatovic and Urusov (2012a) for a pointer to the literature). The new proof has an intuitive interpretation as we link the integral functional to the explosion time of an associated diffusion process.

“Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth

We obtain a representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. As in the classical Ray-Knight representation, the excursions of $H$ are the exploration paths of the trees of descendants of the ancestors at time $t=0$, and the local time of $H$ at height $t$ measures the population size at time $t$ (see e.g. \cite{LG4}). We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time $s$ and living at time $t=H_s$ is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating $H$ with a sequence of Harris paths $H^N$ which figure in a Ray-Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of $H^N$ together with its local times {\em and} with the Girsanov densities that introduce the dependence in the reproduction.

The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof

A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray–Knight theorem.