Strong law of large numbers for fragmentation processes

Harris, Simon C.; Knobloch, Robert; Kyprianou, Andreas E.

doi:10.1214/09-aihp311

Cited by 13 publications

(41 citation statements)

References 26 publications

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“…where Ext c is the event that the system does not become extinct, |X t | denotes the number of particles at time t and f is a bounded, continuous function. This is indeed the case as follows from [17]. In Theorem 3.1 we obtain (3) for a slightly more general class of functions.…”

Section: Introductionsupporting

confidence: 58%

CLT for Ornstein-Uhlenbeck branching particle system

Adamczak

Miłoś

2015

Electron. J. Probab.

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In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in R d and undergoing a binary, supercritical branching with a constant rate λ > 0. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. The limit and the CLT normalisation fall into three qualitatively different classes. In, what we call, the small branching rate case the situation resembles the classical one. The weak limit is Gaussian and normalisation is the square root of the size of the system. In the critical case the limit is still Gaussian, however the normalisation requires an additional term. Finally, when branching has large rate the situation is completely different. The limit is no longer Gaussian, the normalisation is substantially larger than the classical one and the convergence holds in probability.We prove also that the spatial fluctuations are asymptotically independent of the fluctuations of the total number of particles (which is a Galton-Watson process). MSC: primary 60F05; 60J80 secondary 60G20

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Section: Introductionsupporting

confidence: 58%

CLT for Ornstein-Uhlenbeck branching particle system

Adamczak

Miłoś

2015

Electron. J. Probab.

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“…For example, they have since been extended to more general branching diffusions in Engländer et al [14] and to fragmentation processes in Harris et al [31]. More classical techniques based on the expectation semigroup are simply not able to generalize easily, since they often require either some a priori bounds on the semigroup or involve difficult estimates -for example, in Harris and Williams [28] their important bound of a non-linear term is made possible only by the existence of a good L 2 theory for their operator, and this is not generally available.…”

Section: Introductionmentioning

confidence: 99%

A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales

Hardy

Harris

2009

Lecture Notes in Mathematics

153

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Summary. We present a modified formalization of the 'spine' change of measure approach for branching diffusions in the spirit of those found in Kyprianou [40] and Lyons et al. [44,43,41]. We use our formulation to interpret certain 'GibbsBoltzmann' weightings of particles and use this to give an intuitive proof of a general 'Many-to-One' result which enables expectations of sums over particles in the branching diffusion to be calculated purely in terms of an expectation of one 'spine' particle. We also exemplify spine proofs of the L p -convergence (p ≥ 1) of some key 'additive' martingales for three distinct models of branching diffusions, including new results for a multi-type branching Brownian motion and discussion of left-most particle speeds.

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“…where t (n) 2 is defined as above. Bearing in mind that |g| ≤ 1 it follows from the DCT in conjunction with (18) and (23), respectively, that which completes the proof, since t ∈ C u(·,x) .…”

Section: Proof Of Propositionmentioning

confidence: 63%

“…holds for each h > 0, where conditionally on F h theÑ (·) t and B (n) are independent copies of N (·) t and B n , respectively. Furthermore, note that (18)…”

Section: Proof Of Propositionmentioning

confidence: 99%

One-Sided FKPP Travelling Waves for Homogeneous Fragmentation Processes

Knobloch

2016

J Theor Probab

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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.2010 Mathematics Subject Classification: 60G09, 60J25.In the context of homogeneous fragmentation processes we prove the existence and uniqueness of one-sided travelling waves within a certain range of wave speeds. More precisely, the problem we are concerned with in this paper can be roughly described as follows. Consider the integro-differential equationfor certain c ∈ R + := (0, ∞) and all x ∈ R + 0 := [0, ∞), where the product is taken over all n ∈ N with |π n | ∈ R + . Here the space P is the space of partitions (π n ) n∈N of N and µ is the so-called dislocation measure on P. This notation is introduced in more detail in the next section. We are interested in solutions f : R → [0, 1] of the above equation that satisfy

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Strong law of large numbers for fragmentation processes

Cited by 13 publications

References 26 publications

CLT for Ornstein-Uhlenbeck branching particle system

CLT for Ornstein-Uhlenbeck branching particle system

A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales

One-Sided FKPP Travelling Waves for Homogeneous Fragmentation Processes

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