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Growth-fragmentation process embedded in a planar Brownian excursion

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The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane \({\mathbb {H}}\), obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski in (Bertoin et al. Probab Theory Relat Field 172:663–724, 2018).

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We are grateful to Jean Bertoin and Bastien Mallein for stimulating discussions, and to Juan Carlos Pardo for a number of helpful discussions regarding self-similar processes. After a first version of this article appeared online, Nicolas Curien pointed to us the connection with random planar maps, and the link between the duration of the excursion and the area of the map. We warmly thank him for his explanations. We also learnt that Timothy Budd in an unpublished note had already predicted the link between growth-fragmentations of [5] and planar excursions. Finally, we thank two anonymous referees for their careful reading and valuable comments.

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Correspondence to William Da Silva.

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Aïdékon, E., Da Silva, W. Growth-fragmentation process embedded in a planar Brownian excursion. Probab. Theory Relat. Fields 183, 125–166 (2022).

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