In the spirit of a classical results for Crump-Mode-Jagers processes, we prove a strong law of large numbers for homogenous fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than $η$ for $1\geq η>0$.
The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes.
In this paper we prove a strong law of large numbers and its L^1-convergence counterpart for the process counted with a random characteristic in the context of self-similar fragmentation processes. This result extends a somewhat analogical result by Nerman for general branching processes to fragmentation processes. In addition, we apply the general result of this paper to a specific example that in particular extends a limit theorem, concerning the fragmentation energy, by Bertoin and Martínez from L^1-convergence to almost sure convergence. Our approach treats fragmentation processes with an …
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.
This text surveys different probabilistic aspects of a model which is used to describe the evolution of an object that falls apart randomly as time passes. Each point of view yields useful techniques to establish properties of such random fragmentation processes.
Growth-fragmentation processes describe systems of particles in which each particle may grow larger or smaller, and divide into smaller ones as time proceeds. Unlike previous studies, which have focused mainly on the self-similar case, we introduce a new type of growth-fragmentation which is closely related to Lévy driven Ornstein-Uhlenbeck type processes. Our model can be viewed as a generalization of compensated fragmentation processes introduced by Bertoin, or the stochastic counterpart of a family of growth-fragmentation equations. We establish a convergence criterion for a sequence of suc…
We define and study a family of Markov processes with state space the compact set of all partitions of N that we call exchangeable fragmentation-coalescence processes. They can be viewed as a combination of exchangeable fragmentation as defined by Bertoin and of homogenous coalescence as defined by Pitman and Schweinsberg or Mohle and Sagitov. We show that they admit a unique invariant probability measure and we study some properties of their paths and of their equilibrium measure.
We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non- random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit self-organised criticality in the cluster size distribution, with …
Markovian growth-fragmentation processes introduced by Bertoin model a system of growing and splitting cells in which the size of a typical cell evolves as a Markov process $X$ without positive jumps. We find that two growth-fragmentation processes associated respectively with two processes $X$ and $Y$ (with different laws) may have the same distribution, if $(X,Y)$ is a bifurcator, roughly speaking, which means that they coincide up to a bifurcation time and then evolve independently. Using this criterion, we deduce that the law of a self-similar growth-fragmentation is determined by a cumula…
We give a stochastic model for the fragmentation phase of a snow avalanche. We construct a fragmentation-branching process related to the avalanches, on the set of all fragmentation sizes introduced by J. Bertoin. A fractal property of this process is emphasized. We also establish a specific stochastic equation of fragmentation. It turns out that specific branching Markov processes on finite configurations of particles with sizes bigger than a strictly positive threshold are convenient for describing the continuous time evolution of the number of the resulting fragments. The results are obtain…
We show that for $0<α<1$ and $θ>-α$, the Poisson-Dirichlet distribution with parameter $(α, θ)$ is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split and merge transformations and the parameter $α=0$.
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 present a first study on the energy required to reduce a unit mass fragment by consecutively using several devices, as it happens in the mining industry. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced by Bertoin and Martinez, we compute the average energy required to attain a size x with this two-device procedure. We then asymptotically compare, as x goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that for certain range of paramet…
We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth-fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.
We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy-Khinchin representation is obtained, using techniques from positive self-similar Mar…
In this paper we extend two limit theorems which were recently obtained for fragmentation processes to such processes with immigration. More precisely, in the setting with immigration we consider a limit theorem for the process counted with a random characteristic as well as the asymptotic behaviour of an empirical measure associated with the stopping line corresponding to the first blocks, in their respective line of descent, that are smaller than a given size. In addition, we determine the asymptotic decay rate of the size of the largest block in a homogeneous fragmentation process with immi…
We consider the height process of a Levy process with no negative jumps, and its associated continuous tree representation. Using Levy snake tools developed by Duquesne and Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is interesting in itself.
In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein-Uhlenbeck process associated with Bertoin's tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Levy processes.
We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using tools developed by Duquesne and Le Gall, we construct a fragmentation process at height, which generalizes the fragmentation at height of stable trees given by Miermont. In this more general framework, we recover that the dislocation measures are the same as the dislocation measures of the fragmentation at node introduced by Abraham and Delmas, up to a factor equal to the fragment size. We also compute the asymptotic for the number of small fragments.
We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let $F\_{1}^{(m)}(t),F\_{2}^{(m)}(t),...$ denote the decreasing rearrangement of the masses present at time $t$ in a such process, starting from an initial mass $m$. Let then $m\to \infty $. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence $(F\_{2}^{(m)},F\_{3}^{(m)},...)$ converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly w…
We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes, simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters $θ_{\star}\leq θ^{\star}\in…
We consider the model of Brownian motion indexed by the Brownian tree. For every $r\geq 0$ and every connected component of the set of points where Brownian motion is greater than $r$, we define the boundary size of this component, and we then show that the collection of these boundary sizes evolves when $r$ varies like a well-identified growth-fragmentation process. We then prove that the same growth-fragmentation process appears when slicing a Brownian disk at height $r$ and considering the perimeters of the resulting connected components.
We provide a new geometric representation of a family of fragmentation processes by nested laminations, which are compact subsets of the unit disk made of noncrossing chords. We specifically consider a fragmentation obtained by cutting a random stable tree at random points, which split the tree into smaller subtrees. When coding each of these cutpoints by a chord in the unit disk, we separate the disk into smaller connected components, corresponding to the smaller subtrees of the initial tree. This geometric point of view allows us in particular to highlight a new relation between the Aldous-P…
Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in a work of Bertoin, where special attention was given to the self-similar case. A Malthusian condition was notably given under which the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. Our main result in this work states the converse: when this condition is not verified, then the growth-fragmentation process explodes alm…
Markovian growth-fragmentation processes introduced by Bertoin extend the pure fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of the known asymptotic behaviors of self-similar pure fragmentations when growth is added to the fragments is a natural question that we investigate in this paper. Our results involve the terminal value of some additive martingales whose uniform integrability is an essential requirement. Dwelling first on the homogeneous case, we exploit the connection with branching random walks and in particular the ma…
A Markov Additive Process is a bi-variate Markov process $(ξ,J)=\big((ξ_t,J_t),t\geq0\big)$ which should be thought of as a multi-type Lévy process: the second component $J$ is a Markov chain on a finite space $\{1,\ldots,K\}$, and the first component $ξ$ behaves locally as a Lévy process, with local dynamics depending on $J$. In the subordinator-like case where $ξ$ is nondecreasing, we establish several results concerning the moments of $ξ$ and of its exponential functional $I_ξ=\int_{0}^{\infty} e^{-ξ_t}\mathrm dt,$ extending the work of Carmona et al., and Bertoin and Yor. We then apply the…
We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time t is encoded by a partition $Π$(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $Π$ is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, r fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent …
In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time behavior of a ranked fragmentation.
Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize $(α,θ)$} and $(β,θ/α)$, wherein $PD(α, θ)$ is coagulated by $PD(β,θ/α)$ for $0<α<1$, $0 \leqβ<1$ and $-β<θ/α$. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of $PD(α,θ)$ coagulated wit…
The Brownian triangulation is a random compact subset of the unit disk introduced by Aldous. For $ε>0$, let $N(ε)$ be the number of triangles whose sizes (measured in different ways) are greater than $ε$ in the Brownian triangulation. We determine the asymptotic behaviour of $N(ε)$ as $ε\to 0$. To obtain this result, a novel concept of "large" dislocations in fragmentations has been proposed. We develop an approach to study the number of large dislocations which is widely applicable to general self-similar fragmentation processes. This technique enables us to study $N(ε)$ because of a bijectio…
We consider the exchangeable fragmentation-coagulation (EFC) processes, where the coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and the fragmentations dislocate at finite rate an individual block into sub-blocks of infinite size. Sufficient conditions are found for the block-counting process to explode (i.e. to reach $\infty$) or not and for infinity to be an exit boundary or an entrance boundary. In a case of regularly varying fragmentation and coagulation mechanisms, we find regimes where the boundary $\infty$ can be either an exit, an entrance or a regular boundary…
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.
We study infinite systems of particles which undergo coalescence and fragmentation, in a manner determined solely by their masses. A pair of particles having masses $x$ and $y$ coalesces at a given rate $K(x,y)$. A particle of mass $x$ fragments into a collection of particles of masses $θ\_1 x, θ\_2 x, \ldots$ at rate $F(x) β(dθ)$. We assume that the kernels $K$ and $F$ satisfy Hölder regularity conditions with indices $λ\in (0,1]$ and $α\in [0, \infty)$ respectively. We show existence of such infinite particle systems as strong Markov processes taking values in $\ell\_λ$, the set of ordered s…
We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Continuum Random Tree is encoded in a normalized Brownian excursion. Under mild hypotheses, we then compute the Hausdorff dimensions of these trees, and the maximal Hölder exponents of the height functions.
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…
Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the b…
We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $β_m$ (if the sampled parts are distinct) or splitting the part with probability $β_s$ according to a law $σ$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $σ$ is the uniform measure then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of ``analytic'' invarian…
Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic behaviour at large times is interesting both mathematically and practically. The spine decomposition is a key tool in its study. In this work, we consider the class of compensated fragmentations, or homogeneous growth-fragmentations, recently defined by Bertoin. We give a complete spine decomposition of these processes in terms of a Lévy process with immigra…
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…
The fragmentation processes of exchangeable partitions have already been studied by several authors. In this paper, we examine rather fragmentation of exchangeable compositions, that means partitions of $\mathbb{N}$ where the order of the blocks counts. We will prove that such a fragmentation is bijectively associated to an interval fragmentation. Using this correspondence, we then calculate the Hausdorff dimension of certain random closed set that arise in interval fragmentations and we study Ruelle's interval fragmentation.
We establish the central limit theorem for the number of groups at the equilibrium of a coagulation-fragmentation process given by a parameter function with polynomial rate of growth. The result obtained is compared with the one for random combinatorial structures obeying the logarithmic condition.
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.
This paper introduces stochastic processes that describe the evolution of systems of particles in which particles immigrate according to a Poisson measure and split according to a self-similar fragmentation. Criteria for existence and absence of stationary distributions are established and uniqueness is proved. Also, convergence rates to the stationary distribution are given. Linear equations which are the deterministic counterparts of fragmentation with immigration processes are next considered. As in the stochastic case, existence and uniqueness of solutions, as well as existence and uniquen…
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.
Similarly as in (Blancas et al. 2018) where nested coalescent processes are studied, we generalize the definition of partition-valued homogeneous Markov fragmentation processes to the setting of nested partitions, i.e. pairs of partitions $(ζ,ξ)$ where $ζ$ is finer than $ξ$. As in the classical univariate setting, under exchangeability and branching assumptions, we characterize the jump measure of nested fragmentation processes, in terms of erosion coefficients and dislocation measures. Among the possible jumps of a nested fragmentation, three forms of erosion and two forms of dislocation are …
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…
We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [dgg] for the above class of processes. The method used goes back to A.Khintchine.
We develop a unified approach to the problem of clustering in the three different fields of applications, as indicated in the title the paper. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fawler method in statistical physics. Our main result is the derivation of asymptotic formulae for probabilities of clusters (= groups) of certain sizes as the number of particles goes to infinity. Based on these formulae we prove the zero-one law for the distribution of the largest cluster and establish the threshold function in the phase transition from 0 to 1 in th…
The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the large time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via nonincreasing self-similar Markov processes that continuously reach 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on n…
