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preprint2016arXiv

Genealogies in Expanding Populations

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [25]. The first step, which is easily accomplished using results of Mueller and Tribe [38], is to show that when space and time are rescaled correctly, our biased voter model converges to a Wright-Fisher SPDE. A simple extension of a result of Durrett and Restrepo [18] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. Brunet et al. [8] have conjectured that genealogies in models of this type are described by the Bolthausen-Sznitman coalescent, see [39]. However, in the model we study there are no simultaneous coalescences. Our third and most significant result concerns "tracer dynamics" in which some of the initial particles in the biased voter model are labeled. We show that the joint distribution of the labeled and unlabeled particles converges to the solution of a system of stochastic partial differential equations. A new duality equation that generalizes the one Shiga [44] developed for the Wright-Fisher SPDE is the key to the proof of that result.

preprint2015arXiv

On stochastic comparisons of largest order statistics in the scale model

Let $X_{λ_{1}},X_{λ_{2}},\ldots ,X_{λ_{n}}$ be independent nonnegative random variables with $X_{λ_{i}}\sim F(λ_{i}t)$, $i=1,\ldots ,n$, where $λ_{i}>0$, $i=1,\ldots ,n$ and $F$ is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic $X_{n:n}^{λ}$ is smaller than another one $X_{n:n}^{θ}$ according to likelihood ratio ordering. Furthermore, we apply these results when $F$ is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases.

preprint2015arXiv

Fluctuation limit for interacting diffusions with partial annihilations through membranes

We study fluctuations of the empirical processes of a non-equilibrium interacting particle system consisting of two species over a domain that is recently introduced in [8] and establish its functional central limit theorem. This fluctuation limit is a distribution-valued Gaussian Markov process which can be represented as a mild solution of a stochastic partial differential equation. The drift of our fluctuation limit involves a new partial differential equation with nonlinear coupled term on the interface that characterized the hydrodynamic limit of the system. The covariance structure of the Gaussian part consists two parts, one involving the spatial motion of the particles inside the domain and other involving a boundary integral term that captures the boundary interactions between two species. The key is to show that the Boltzmann-Gibbs principle holds for our non-equilibrium system. Our proof relies on generalizing the usual correlation functions to the join correlations at two different times.

preprint2016arXiv

Asymptotic analysis of symmetric functions

In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed.

preprint2015arXiv

Boundary values, random walks and $\ell^p$-cohomology in degree one

The vanishing of reduced $\ell^2$-cohomology for amenable groups can be traced to the work of Cheeger & Gromov. The subject matter here is reduced $\ell^p$-cohomology for $p \in ]1,\infty[$, particularly its vanishing. Results showing its triviality are obtained, for example: when $p \in ]1,2]$ and $G$ is amenable; when $p \in ]1,\infty[$ and $G$ is Liouville (in particular, of intermediate growth). This is done by answering a question of Pansu assuming the graph satisfies an isoperimetric profile. Namely, the triviality of the reduced $\ell^p$-cohomology is equivalent to the absence of non-constant bounded (equivalently, not necessarily bounded) harmonic functions with gradient in $\ell^q$ ($q$ depends on the profile). In particular, one reduces questions of non-linear analysis ($p$-harmonic functions) to linear ones (harmonic functions with a restrictive growth condition).

preprint2012arXiv

Hitting distributions of alpha-stable processes via path censoring and self-similarity

In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of a type of path censoring. Second, we are able to describe in explicit analytical detail a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process from which the desired solution can be sourced. Moreover, as a consequence of this approach, we are able to deliver a number of additional, related identities in explicit form for alpha-stable processes.

preprint2012arXiv

On the existence of non-central Wishart distributions

This paper deals with the existence issue of non-central Wishart distributions which is a research topic initiated by Wishart (1928), and with important contributions by e.g., Lévy (1937), Gindikin (1975), Shanbhag (1988), Peddada and Richards (1991). We present a new method involving the theory of affine Markov processes, which reveals joint necessary conditions on shape and non-centrality parameter. While Eaton's conjecture concerning the necessary range of the shape parameter is confirmed, we also observe that it is not sufficient anymore that it only belongs to the Gindikin ensemble, as is in the central case.

preprint2014arXiv

Hydrodynamic Limits and Propagation of Chaos for Interacting Random Walks in Domains

A new non-conservative stochastic reaction-diffusion system in which two families of random walks in two adjacent domains interact near the interface is introduced and studied in this paper. Such a system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. We show that in the macroscopic limit, the particle densities converge to the solution of a coupled nonlinear heat equations. For this, we first prove that propagation of chaos holds by establishing the uniqueness of a new BBGKY hierarchy. A local central limit theorem for reflected diffusions in bounded Lipschitz domains is also established as a crucial tool.

preprint2013arXiv

Potentials of stable processes

For a stable process, we give an explicit formula for the potential measure of the process killed outside a bounded interval and the joint law of the overshoot, undershoot and undershoot from the maximum at exit from a bounded interval. We obtain the equivalent quantities for a stable process reflected in its infimum. The results are obtained by exploiting a simple connection with the Lamperti representation and exit problems of stable processes.

preprint2014arXiv

Functional central limit theorem for Brownian particles in domains with Robin boundary condition

We rigorously derive non-equilibrium space-time fluctuation for the particle density of a system of reflected diffusions in bounded Lipschitz domains in $\mathbb R^d$. The particles are independent and are killed by a time-dependent potential which is asymptotically proportional to the boundary local time. We generalize the functional analytic framework introduced by Kotelenez [19, 20] to deal with time-dependent perturbations. Our proof relies on Dirichlet form method rather than the machineries derived from Kotelenez's sub-martingale inequality. Our result holds for any symmetric reflected diffusion, for any bounded Lipschitz domain and for any dimension $d\geq 1$.

preprint2016arXiv

Discrete approximations to local times for reflected diffusions

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [7]. Applications of the join convergence result to PDE problems are illustrated.

preprint2016arXiv

The Surface Area Deviation of the Euclidean Ball and a Polytope

While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices in the symmetric surface area deviation.

preprint2016arXiv

Diffusion of innovation in large scale graphs

Will a new smartphone application diffuse deeply in the population or will it sink into oblivion soon? To predict this, we argue that common models of spread of innovations based on cascade dynamics or epidemics may not be fully adequate. Therefore we propose a novel stochastic network dynamics modeling the spread of a new technological asset, whose adoption is based on the word-of-mouth and the persuasion strength increases the more the product is diffused. In this paper we carry on an analysis on large scale graphs to show off how the parameters of the model, the topology of the graph and, possibly, the initial diffusion of the asset, determine whether the spread of the asset is successful or not. In particular, by means of stochastic dominations and deterministic approximations, we provide some general results for a large class of expansive graphs. Finally we present numerical simulations trying to expand the analytical results we proved to even more general topologies.

preprint2015arXiv

Probabilistic aspects of critical growth-fragmentation equations

The self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered by Doumic and Escobedo in the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the non-homogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter $(0,\infty)$ continuously from either $0$ or $\infty$, we exhibit unexpected spontaneous generation of mass in the solutions.

preprint2015arXiv

Single-Seed Cascades on Clustered Networks

We consider a dynamic network cascade process developed by Watts applied to a random networks with a specified amount of clustering, belonging to a class of random networks developed by Newman. We adapt existing tree-based methods to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node, and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a special case of a formula of Hackett et al. giving conditions for certain extinction of the cascade.

preprint2014arXiv

The extended hypergeometric class of Lévy processes

With a view to computing fluctuation identities related to stable processes, we review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (arXiv:1012.0817). We give the Wiener-Hopf factorisation of a process in the extended class, and characterise its exponential functional. Finally, we give three concrete examples arising from transformations of stable processes.

preprint2012arXiv

Wishart Processes and Wishart Distributions: An Affine Processes Point of View

These lecture notes provide an introduction to the theory of Wishart distributions and Wishart processes, particularly what concerns their existence and their realizations as solutions of Wishart SDEs. The material is based on recent developments in the theory of affine processes and related fields.

preprint2017arXiv

Blocking duality for $p$-modulus on networks and applications

This paper explores the implications of blocking duality---pioneered by Fulkerson et al.---in the context of $p$-modulus on networks. Fulkerson's blocking duality is an analogue on networks to the method of conjugate families of curves in the plane. The technique presented here leads to a general framework for studying families of objects on networks; each such family has a corresponding dual family whose $p$-modulus is essentially the reciprocal of the original family's. As an application, we give a modulus-based proof for the fact that effective resistance is a metric on graphs. This proof immediately generalizes to yield a family of graph metrics, depending on the parameter $p$, that continuously interpolates among the shortest-path metric, the effective resistance metric, and the mincut ultrametric. In a second application, we establish a connection between Fulkerson's blocking duality and the probabilistic interpretation of modulus. This connection, in turn, provides a straightforward proof of several monotonicity properties of modulus that generalize known monotonicity properties of effective resistance. Finally, we use this framework to expand on a result of Lovász in the context of randomly weighted graphs.

preprint2018arXiv

Mass-structure of weighted real trees

Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is a distinguished root, and $p$ is a probability measure on this space. The underlying branching structure is carried implicitly in the metric $d$. We explore various ways of describing the interaction between branching structure and mass in $(\mathcal{T},d,r,p)$ in a way that depends on $d$ only by way of this branching structure. We introduce a notion of mass-structure equivalence and show that two rooted, weighted $\mathbb{R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingman's paintbox, have the same distribution. We introduce a family of trees, called "interval partition trees" that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.

preprint2016arXiv

Minimal subfamilies and the probabilistic interpretation for modulus on graphs

The notion of $p$-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for $p$-modulus. We show that minimal subfamilies have at most $|E|$ elements and that these elements carry a weight related to their "importance" in relation to the corresponding $p$-modulus problem. When $p=2$, this measure of importance is in fact a probability measure and modulus can be thought as trying to minimize the expected overlap in the family.

preprint2017arXiv

A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymp-totic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach to the study of this asymptotic behaviour. We use a Feynman--Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the spectral radius and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual. In special cases, we obtain exponential convergence.

preprint2018arXiv

Tail and moment estimates for a class of random chaoses of order two

We derive two-sided bounds for moments and tails of random quadratic forms (random chaoses of order $2$), generated by independent symmetric random variables such that $\lVert X \rVert_{2p} \leq α\lVert X \rVert_p$ for any $p\geq 1$ and some $α\geq 1$. Estimates are deterministic and exact up to some multiplicative constants which depend only on $α$.

preprint2017arXiv

Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations

Semigroups, generated by Feller processes killed upon leaving a given domain, are considered. These semigroups correspond to Cauchy-Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation converts into a Feynman formula, i.e. into a limit of $n$-fold iterated integrals of certain functions as $n\to\infty$. Representations of solutions of evolution equations by Feynman formulae can be used for direct calculations and simulation of underlying stochasstic processes. Further, a method to approximate solutions of time-fractional (including distributed order time-fractional) evolution equations is suggested. This method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. Moreover, this method leads to Feynman formulae for solutions of time-fractional evolution equations. To illustrate the method, a class of distributed order time-fractional diffusion equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy-Dirichlet problems are obtained.

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math.PR

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24 featured work(s)

Genealogies in Expanding Populations

On stochastic comparisons of largest order statistics in the scale model

Fluctuation limit for interacting diffusions with partial annihilations through membranes

Asymptotic analysis of symmetric functions

Boundary values, random walks and $\ell^p$-cohomology in degree one

Hitting distributions of alpha-stable processes via path censoring and self-similarity

On the existence of non-central Wishart distributions

Hydrodynamic Limits and Propagation of Chaos for Interacting Random Walks in Domains

Potentials of stable processes

Functional central limit theorem for Brownian particles in domains with Robin boundary condition

Discrete approximations to local times for reflected diffusions

The Surface Area Deviation of the Euclidean Ball and a Polytope

Diffusion of innovation in large scale graphs

Probabilistic aspects of critical growth-fragmentation equations

Single-Seed Cascades on Clustered Networks

The extended hypergeometric class of Lévy processes

Wishart Processes and Wishart Distributions: An Affine Processes Point of View

Blocking duality for $p$-modulus on networks and applications

Mass-structure of weighted real trees

Minimal subfamilies and the probabilistic interpretation for modulus on graphs

A probabilistic approach to spectral analysis of growth-fragmentation equations

Tail and moment estimates for a class of random chaoses of order two

Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations

On a class of Time-fractional Continuous-state Branching Processes

12 visible researcher(s)

Hao Wang

Meng Wang

Yang Li

Yu Zhang

Rui Zhang

Chang Liu

Lei Li

J. Wang

Jian Wang

Jing Zhang

Fan Yang

Wei Liu