Survival of homogeneous fragmentation processes with killing

Knobloch, Robert; Kyprianou, Andreas E.

doi:10.1214/12-aihp520

Cited by 2 publications

(14 citation statements)

References 19 publications

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“…Analysis of this martingale in [56], in particular making use of known properties of scale functions, allows the authors to deduce that, under mild conditions, there exists a unique constant p * > 0 such that whenever c > φ (p * )…”

Section: Scale Functions and Applied Probabilitymentioning

confidence: 99%

The Theory of Scale Functions for Spectrally Negative Lévy Processes

Kuznetsov

Kyprianou

Rivero

2012

Lecture Notes in Mathematics

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249

363

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The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy-Khintchine formula and its relationship to the Lévy-Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes;

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Section: Scale Functions and Applied Probabilitymentioning

confidence: 99%

The Theory of Scale Functions for Spectrally Negative Lévy Processes

Kuznetsov

Kyprianou

Rivero

2012

Lecture Notes in Mathematics

Self Cite

249

363

View full text Add to dashboard Cite

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“…Throughout this paper we assume that d = 0 as well as ν(s ∈ S : s 2 = 0) = 0. In view of the forthcoming assumption (4) this enables us to resort to the results of [19], where the same assumptions are made. Let us mention that the assumption d = 0 does not result in any loss of generality, see Remark 1.…”

Section: Homogeneous Fragmentation Processes With Killingmentioning

confidence: 99%

“…For the time being, let x ∈ R ∞ . In this paper we are concerned with a specific procedure of killing blocks of Π, see Figure 1(a), that was introduced in [19]. More precisely, for c > 0 a block Π n (t) is killed, with cemetery state ∅, at the moment of its creation t ∈ R + 0 if |Π n (t)| < e −(x+ct) .…”

Section: Homogeneous Fragmentation Processes With Killingmentioning

confidence: 99%

“…ζ x is the supremum of all the killing times of individual blocks. The question whether ζ x is finite or infinite was considered in Theorem 2 of [19], see also Proposion 2 below. Furthermore, define a function ϕ : R → [0, 1] by…”

Section: Homogeneous Fragmentation Processes With Killingmentioning

confidence: 99%

“…Our approach is based on using fragmentation processes with killing at an exponential barrier. These processes have been studied in [19] and we briefly describe the corresponding concepts below.…”

Section: Introductionmentioning

confidence: 99%

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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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Survival of homogeneous fragmentation processes with killing

Cited by 2 publications

References 19 publications

The Theory of Scale Functions for Spectrally Negative Lévy Processes

The Theory of Scale Functions for Spectrally Negative Lévy Processes

One-Sided FKPP Travelling Waves for Homogeneous Fragmentation Processes

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