Changing the branching mechanism of a continuous state branching process using immigration

Abraham, Romain; Delmas, Jean‐François

doi:10.1214/07-aihp165

Cited by 9 publications

(68 citation statements)

References 15 publications

(45 reference statements)

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“…Next, let us consider the case σ 2 +b > 0. When σ 2 = 0 and b > 0 the statement follows from the preceding discussion as lim y→0 κ(y) ≥ lim y→0 bW (1) (2). Assume first that σ = 0, then necessarily b = 0 as otherwise κ = ∞.…”

Section: Derivatives and Smoothness Of Convolutionsmentioning

confidence: 83%

“…(2.12) below, given in Proposition 1.6 in combination with Lemma 2.5 below, upon imposing higher order continuous differentiability on W (1) andμ. If σ > 0, the problem of higher order (non-)differentiability of W (1) is studied in Chan et al [9]. In particular, Theorem 2 in [9] says that if the Blumenthal-Getoor lower index inf{β > 0; 1 0 r β Π(dr) < ∞} < 2, then W (2) ∈ C n+1 (R + ) if and only if Π ∈ C n (R + ).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Hence, in both cases, h ′ is in C(R + ), which implies by the fundamental theorem of calculus that h is (continuously) differentiable on R + with derivative given by h (1)…”

Section: Derivatives and Smoothness Of Convolutionsmentioning

confidence: 98%

“…Now assume that g is in C(R + ) and let y > 0. If f ∈ C 1 (R + ), then f (1) and g are bounded on sets of the form [a, b] ⊂ R + and therefore by the dominated convergence theorem,…”

Section: Derivatives and Smoothness Of Convolutionsmentioning

confidence: 99%

“…We mention that Schilling and Wang in [45] provide interesting sufficient conditions on the transition kernel of Feller semigroups which imply the strong Feller property. However, we have not been able to use their criteria and instead we prove this property directly using Theorem 1.1 (1). To this end, let t > T 0 and f ∈ B b (R + ≥0 ).…”

Section: 12mentioning

confidence: 99%

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Smoothness of continuous state branching with immigration semigroups

Chazal

Loeffen

Patie

2018

Journal of Mathematical Analysis and Applications

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In this work we develop an original and thorough analysis of the (non)-smoothness properties of the semigroups, and their heat kernels, associated to a large class of continuous state branching processes with immigration. Our approach is based on an in-depth analysis of the regularity of the absolutely continuous part of the invariant measure combined with a substantial refinement of Ogura's spectral expansion of the transition kernels. In particular, we find new representations for the eigenfunctions and eigenmeasures that allow us to derive delicate uniform bounds that are useful for establishing the uniform convergence of the spectral representation of the semigroup acting on linear spaces that we identify. We detail several examples which illustrate the variety of smoothness properties that CBI transition kernels may enjoy and also reveal that our results are sharp. Finally, our technique enables us to provide the (eventually) strong Feller property as well as the rate of convergence to equilibrium in the total variation norm. ∞ 0 Π(dy) = ∞ and where A = D + L is purely non-local, that is D ≡ 0.

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Section: Derivatives and Smoothness Of Convolutionsmentioning

confidence: 83%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Hence, in both cases, h ′ is in C(R + ), which implies by the fundamental theorem of calculus that h is (continuously) differentiable on R + with derivative given by h (1)…”

Section: Derivatives and Smoothness Of Convolutionsmentioning

confidence: 98%

“…Now assume that g is in C(R + ) and let y > 0. If f ∈ C 1 (R + ), then f (1) and g are bounded on sets of the form [a, b] ⊂ R + and therefore by the dominated convergence theorem,…”

Section: Derivatives and Smoothness Of Convolutionsmentioning

confidence: 99%

Section: 12mentioning

confidence: 99%

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Smoothness of continuous state branching with immigration semigroups

Chazal

Loeffen

Patie

2018

Journal of Mathematical Analysis and Applications

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Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations

Abraham

Delmas

2009

Stochastic Processes and their Applications

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112

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We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams' decomposition of the genealogy of the total population given by a continuum random tree, according to the ancestral lineage of the last individual alive. This allows us to give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.

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A continuum-tree-valued Markov process

Abraham¹,

Delmas²

2012

Ann. Probab.

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137

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We present a construction of a L\'evy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $\psi$ be a critical branching mechanism. We set $\psi_\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let $\Theta=(\theta_\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set of values of $\theta$ for which $\psi_\theta$ is a branching mechanism. The pruning procedure allows to construct a decreasing L\'evy-CRT-valued Markov process $(\ct_\theta,\theta\in\Theta)$, such that $\mathcal{T}_\theta$ has branching mechanism $\psi_\theta$. It is sub-critical if $\theta>0$ and super-critical if $\theta<0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $\theta$ for which $\mathcal{T}_\theta$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT

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Changing the branching mechanism of a continuous state branching process using immigration

Cited by 9 publications

References 15 publications

Smoothness of continuous state branching with immigration semigroups

Smoothness of continuous state branching with immigration semigroups

Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations

A continuum-tree-valued Markov process

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