Matthias Winkel

University of Oxford

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Diffusions on a space of interval partitions: construction from Bertoin’s ${\tt BES}_{0}(d)$, $d\in (0,1)$

Electronic Communications in Probability (2020)

Matthias Winkel

In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable Lévy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.

Bessel process

10.1214/20-ecp355

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Correction: Phase-3 trial of recombinant human alkaline phosphatase for patients with sepsis-associated acute kidney injury (REVIVAL)

Intensive Care Medicine (2024)

Peter Pickkers Derek C. Angus Kristie Bass Rinaldo Bellomo Erik van den Berg

Pain medicine

10.1007/s00134-024-07357-z

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

arXiv (Cornell University) (2018)

Jim Pitman Matthias Winkel

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.

Bessel process

Mean squared displacement

Fractional Brownian motion

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Partly exchangeable fragmentations

Matthias Winkel

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this thesis, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour. We further extend the underlying exchangeable fragmentation processes of such trees into partly exchangeable fragmentation processes by weakening the exchangeability. We obtain the integral representations for the measures associated with partly exchangeable fragmentation processes and subordinator of the tagged fragments. We also embed the trees associated with such processes into continuum random trees and study their limiting behaviour. In the end, we generate a three-parameter family of partly exchangeable trees which contains the family of the alpha-gamma trees and another important two-parameter family based on Poisson-Dirichlet distributions.

Subordinator

Fragmentation

Limiting

Tree (set theory)

Random tree

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Hereditary tree growth and Levy forests

arXiv (Cornell University) (2012)

Thomas Duquesne Matthias Winkel

We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our framework is the space $\bT$ of pointed isometry classes of locally compact rooted real trees equipped with the Gromov-Hausdorff distance. Some of the main results of the paper are a general tightness criterion in $\bT$ and limit theorems for growing families of trees. We apply these results to Galton-Watson trees with exponentially distributed edge lengths. This class is preserved by hereditary reduction. Then we consider families of such Galton-Watson trees that are consistent under hereditary reduction and that we call growth processes. We prove that the associated families of offspring distributions are completely characterised by the branching mechanism of a continuous-state branching process. We also prove that such growth processes converge to Levy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Levy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton-Watson trees, including the super-critical cases.

Branching process

Tree (set theory)

Branching (polymer chemistry)

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Diffusions on a space of interval partitions with Poisson-Dirichlet stationary distributions

arXiv (Cornell University) (2016)

Noah Forman Soumik Pal Douglas Rizzolo Matthias Winkel

We construct a pair of related diffusions on a space of interval partitions of the unit interval $[0,1]$ that are stationary with the Poisson-Dirichlet laws with parameters (1/2,0) and (1/2,1/2) respectively. These are two particular cases of a general construction of such processes obtained by decorating the jumps of a spectrally positive L\'evy process with independent squared Bessel excursions. The processes of ranked interval lengths of our partitions are members of a two parameter family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step towards describing a diffusion on the space of real trees whose existence has been conjectured by Aldous.

Unit interval

10.48550/arxiv.1609.06706

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A recursive distribution equation for the stable tree

arXiv (Cornell University) (2018)

Nicholas Chee Franz Rembart Matthias Winkel

We provide a new characterisation of Duquesne and Le Gall's $α$-stable tree, $α\in(1,2]$, as the solution of a recursive distribution equation (RDE) of the form $\mathcal{T}\overset{d}{=}g(ξ,\mathcal{T}_i, i\geq0)$, where $g$ is a concatenation operator, $ξ= (ξ_i, i\geq 0)$ a sequence of scaling factors, $\mathcal{T}_i$, $i \geq 0$, and $\mathcal{T}$ are i.i.d. trees independent of $ξ$. This generalises a version of the well-known characterisation of the Brownian Continuum Random Tree due to Aldous, Albenque and Goldschmidt. By relating to previous results on a rather different class of RDE, we explore the present RDE and obtain for a large class of similar RDEs that the fixpoint is unique (up to multiplication by a constant) and attractive.

Tree (set theory)

10.48550/arxiv.1812.08636

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Ranked masses in two-parameter Fleming–Viot diffusions

Transactions of the American Mathematical Society (2022)

Noah Forman Soumik Pal Douglas Rizzolo Matthias Winkel

Previous work constructed Fleming–Viot-type measure-valued diffusions (and diffusions on a space of interval partitions of the unit interval[0,1][0,1]) that are stationary with respect to the Poisson–Dirichlet random measures with parametersα∈(0,1)\alpha \in (0,1)andθ>−α\theta > -\alpha. In this paper, we complete the proof that these processes resolve a conjecture by Feng and Sun [Probab. Theory Related Fields 148 (2010), pp. 501–525] by showing that the processes of ranked atom sizes (or of ranked interval lengths) of these diffusions are members of a two-parameter family of diffusions introduced by Petrov [Funct. Anal. Appl. 43 (2009), pp. 279–296], extending a model by Ethier and Kurtz [Adv. in Appl. Probab. 13 (1981), pp. 429–452] in the caseα=0\alpha =0.

10.1090/tran/8764

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Burgers turbulence initialized by a regenerative impulse

Stochastic Processes and their Applications (2001)

Matthias Winkel

Inviscid flow

Particle system

10.1016/s0304-4149(00)00098-3

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Growth of Galton-Watson trees: immigration and lifetimes

arXiv (Cornell University) (2010)

Matthias Winkel

We study certain consistent families (F�)��0 of Galton-Watson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in F�. Specifically, consistency here refers to the property that for each � � �, the forest Fµ has the same distribution as the subforest of Fspanned by the black leaves in a Bernoulli leaf colouring, where each leaf of Fis coloured in black independently with probability �=�. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes. We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, related to Sagitov's (non-Markovian) generalisation of continuous-state branching renewal processes, and similar processes with immigration. AMS 2000 subject classifications: 60J80.

Branching process

Branching (polymer chemistry)

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Citations (3)