Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogs of a certain type of branching random walks, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations.
On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of $\N$. Spatial discretization is especially well-suited to develop directly for continuous times the conceptual method of probability tilting of Lyons, Pemantle and Peres.
We show that in homogeneous fragmentation processes the largest fragment at time $t$ has size $e^{-t Φ'(\bar{p})}t^{-\frac32 (\log Φ)'(\bar{p})+o(1)},$ where $Φ$ is the Lévy exponent of the fragmentation process, and $\bar{p}$ is the unique solution of the equation $(\log Φ)'(\bar{p})=\frac1{1+\bar{p}}$. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.
In this paper we introduce the one-sided FKPP equation in the context of homogeneous fragmentation processes. The main result of the present paper is concerned with the existence and uniqueness of one-sided FKPP travelling waves in this setting. Moreover, we prove some analytic properties of such travelling waves. Our techniques make use of fragmentation processes with killing, an associated product martingale as well as various properties of Lévy processes.
We consider a homogenous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the growth of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of extinction of the killed process.
We consider a mass-conservative fragmentation of the unit interval. The main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of Lévy processes constrained to stay in a finite interval.
A hierarchical system of equations is introduced to describe dynamics of `sizes' of infinite clusters which coagulate and fragmentate with homogeneous rates of certain form. We prove that this system of equations is solved weakly by correlation measures for stochastic dynamics of interval partitions evolving according to some split-merge transformations. Regarding those processes, a sufficient condition for a distribution to be reversible is given. Also, an asymptotic result for properly rescaled processes is shown to obtain a solution to a nonlinear equation called the coagulation-fragmentati…
Homogeneous mass fragmentation processes describe the evolution of a unit mass that breaks down randomly into pieces as time. Mathematically speaking, they can be thought of as continuous-time analogues of branching random walks with non-negative displacements. Following recent developments in the theory of branching random walks, in particular the work of \cite{AShi10}, we consider the problem of the Seneta-Heyde norming of the so-called additive martingale at criticality. Aside from replicating results for branching random walks in the new setting of fragmentation processes, our main goal is…
The main focus of this work is the asymptotic behavior of mass-conservative homogeneous fragmentations. Considering the logarithm of masses makes the situation reminiscent of branching random walks. The standard approach is to study {\bf asymptotical} exponential rates. For fixed $v > 0$, either the number of fragments whose sizes at time $t$ are of order $\e^{-vt}$ is exponentially growing with rate $C(v) > 0$, i.e. the rate is effective, or the probability of presence of such fragments is exponentially decreasing with rate $C(v) < 0$, for some concave function $C$. In a recent paper, N. Krel…
In this paper, we study Ruelle's probability cascades in the framework of time-inhomogeneous fragmentation processes. We describe Ruelle's cascades mechanism exhibiting a family of measures $(ν_t,t\in [0,1[)$ that characterizes its infinitesimal evolution. To this end, we will first extend the time-homogeneous fragmentation theory to the inhomogeneous case. In the last section, we will study the behavior for small and large times of Ruelle's fragmentation process.
Asymptotics for Beta-Splitting Trees via Homogeneous Fragmentations and Meromorphic Potential Theory
Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, Bulgaria（索非亚大学“圣克利门特奥赫里德斯基”数学与信息学院）；Institute of Mathematics, University of Zurich, Switzerland（苏黎世大学数学研究所）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria（保加利亚科学院数学与信息研究所）
We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behaviours of the shattering times, defined as the first instants when all the blocks of the partition process have cardinality less than a fixed integer. Our results may be applied to the study of certain random split trees.
We formulate the notion of the classical Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323-331] and [Comm. Pure Appl. Math. 29 (1976) 553-554], Neveu [In Seminar on Stochastic Processes (1988) 223-242 Birkhäuser] and Chauvin [Ann. Probab. 19 (1991) 1195-1205], our analysis exposes the relation between traveling waves and certa…
A homogeneous mass-fragmentation, as it has been defined in \cite{RFC}, describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a type may represent, for instance, a geometrical shape, and can take finitely many values. In this setting, the dynamics of a randomly tagged fragment play a crucial role in the analysis of the fragmentation. They are determined by a Markov additive process whose distribution depends explicitly on the …
We study the large time behaviour of the mass (size) of particles described by the fragmentation equation with homogeneous breakup kernel. We give necessary and sufficient conditions for the convergence of solutions to the unique self-similar solution.
We establish a characterization of coagulation-fragmentation processes, such that the induced birth and death processes depicting the total number of groups at time $t\ge 0$ are time homogeneous. Based on this, we provide a characterization of mean-field Gibbs coagulation-fragmentation models, which extends the one derived by Hendriks et al. As a by- product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to mean-field models.
For a subcritical Galton-Watson process $(ζ_n)$, it is well known that under an $X \log X$ condition, the quotient $P(ζ_n > 0)/ Eζ_n$ has a finite positive limit. There is an analogous result for a (one-dimensional) supercritical branching random walk: when $a$ is in the so-called subcritical speed area, the probability of presence around $na$ in the $n$-th generation is asymptotically proportional to the corresponding expectation. In Rouault (1993) this result was stated under a natural $X \log X$ assumption on the offspring point process and a (unnatural) condition on the offspring mean. Her…
We consider an infinite balls-in-boxes occupancy scheme with boxes organised in nested hierarchy, and random probabilities of boxes defined in terms of iterated fragmentation of a unit mass. We obtain a multivariate functional limit theorem for the cumulative occupancy counts as the number of balls approaches infinity. In the case of fragmentation driven by a homogeneous residual allocation model our result generalises the functional central limit theorem for the block counts in Ewens' and more general regenerative partitions.
The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo ($2016$) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely \begin{equation} c(x) = \begin{cases} a_{_-} x \quad \quad x < 1 \\ a_{_+} x \quad \quad x \geq 1, \end{cases} \end{equation} with $0 < a_{_+} < a_{_-}$. We giv…
Motivated by nucleation and molecular aggregation in physical, chemical and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistic of times it requires for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volume, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volume, we develop a scaling framework to study the behavior of the fi…
