Growth-fragmentation process embedded in a planar Brownian excursion

Aïdékon, E.; W., Da Silva,

doi:10.48550/arxiv.2005.06372

Cited by 2 publications

(6 citation statements)

References 9 publications

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“…a We may now present an application of Proposition 7.4, which is similar to Proposition 2.7 in [AS20]. We show that, almost surely, excursions cut at heights do not make bubbles above any hyperplane.…”

Section: The Excursion Measure {Sec:excursimentioning

confidence: 73%

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Self-similar growth-fragmentation processes with types

Silva¹,

Pardo²

2021

Preprint

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In this paper, we are interested in self-similar growth-fragmentation processes with types. It roughly consists of two parts. In the first part, we investigate multitype versions of the self-similar growth-fragmentation processes introduced in [Ber17], therefore extending the signed case of [Sil21] to finitely many types. Our main result in this direction describes the law of the spine in the multitype setting. We stress that our arguments only rely on the structure of the underlying Markov additive processes, and hence is more general than [Sil21]. In the second part, we study R d -valued self-similar growth-fragmentation processes driven by an isotropic process. These can be seen as multitype growth-fragmentation processes, where the set of types is the sphere S d−1 . We give the spinal description in this setting. Finally, we prove that such a family of processes shows up when slicing half-space excursions with hyperplanes.

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Section: The Excursion Measure {Sec:excursimentioning

confidence: 73%

“…Proof. We refer to [AS20] for the proof in the planar case, which is easily extended to higher dimensions.…”

Section: The Branching Property Of Excursions In H +mentioning

confidence: 99%

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Self-similar growth-fragmentation processes with types

Silva¹,

Pardo²

2021

Preprint

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“…It is an example of a positive, self-similar Markov process (pssMp). In general, a process is called a pssMp with index θ if it is a standard Markov process with state space [0, ∞), has 0 as an absorbing state, and satisfies the scaling property (1).…”

Section: Ricocheted Stable Processesmentioning

confidence: 99%

“…As it turns out, the same process can be used to describe the excursions of a planar Brownian motion, as explored by [1]. We discuss this in more detail in section 3.…”

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confidence: 98%

A growth-fragmentation model connected to the ricocheted stable process

Watson¹

2021

Preprint

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Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity.

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Growth-fragmentation process embedded in a planar Brownian excursion

Cited by 2 publications

References 9 publications

Self-similar growth-fragmentation processes with types

Self-similar growth-fragmentation processes with types

A growth-fragmentation model connected to the ricocheted stable process

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