The extremal point process of branching brownian motion in Rd
We consider a branching Brownian motion in R d with d ≥ 1 in which the position X (u) t ∈ R d of a particle u at time t can be encoded by its direction θ (u) t ∈ S d−1 and its distance R (u) t to 0. We prove that the extremal point process Pδ (θ (u) t ,R(u) t −m (d) t ) (where the sum is over all pa...
| Main Authors: | , , , , |
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| Format: | Journal article |
| Language: | English |
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Institute of Mathematical Statistics
2024
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| _version_ | 1811139264773095424 |
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| author | Berestycki, J Kim, Y Zeitouni, O Mallein, B Lubetzky, E |
| author_facet | Berestycki, J Kim, Y Zeitouni, O Mallein, B Lubetzky, E |
| author_sort | Berestycki, J |
| collection | OXFORD |
| description | We consider a branching Brownian motion in R d with d ≥ 1 in which the position X (u) t ∈ R d of a particle u at time t can be encoded by its direction θ (u) t ∈ S d−1 and its distance R (u) t to 0. We prove that the extremal point process Pδ (θ (u) t ,R(u) t −m (d) t ) (where the sum is over all particles alive at time t and m (d) t is an explicit centering term) converges in distribution to a randomly shifted, decorated Poisson point process on S d−1 × R. More precisely, the so-called clan-leaders form a Cox process with intensity proportional to D∞(θ)e − √ 2rdrdθ, where D∞(θ) is the limit of the derivative martingale in direction θ and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasinski, Berestycki and Mallein (Ann. Inst. H. ´ Poincaré 57:1786–1810, 2021). The proof builds on that paper and on Kim, Lubetzky and Zeitouni (Ann. Appl. Prob. 33(2):1315–1368, 2023). |
| first_indexed | 2024-03-07T08:10:53Z |
| format | Journal article |
| id | oxford-uuid:46230786-edb8-49af-8571-f382840786ff |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:03:20Z |
| publishDate | 2024 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | oxford-uuid:46230786-edb8-49af-8571-f382840786ff2024-05-17T09:19:03ZThe extremal point process of branching brownian motion in RdJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:46230786-edb8-49af-8571-f382840786ffEnglishSymplectic ElementsInstitute of Mathematical Statistics2024Berestycki, JKim, YZeitouni, OMallein, BLubetzky, EWe consider a branching Brownian motion in R d with d ≥ 1 in which the position X (u) t ∈ R d of a particle u at time t can be encoded by its direction θ (u) t ∈ S d−1 and its distance R (u) t to 0. We prove that the extremal point process Pδ (θ (u) t ,R(u) t −m (d) t ) (where the sum is over all particles alive at time t and m (d) t is an explicit centering term) converges in distribution to a randomly shifted, decorated Poisson point process on S d−1 × R. More precisely, the so-called clan-leaders form a Cox process with intensity proportional to D∞(θ)e − √ 2rdrdθ, where D∞(θ) is the limit of the derivative martingale in direction θ and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasinski, Berestycki and Mallein (Ann. Inst. H. ´ Poincaré 57:1786–1810, 2021). The proof builds on that paper and on Kim, Lubetzky and Zeitouni (Ann. Appl. Prob. 33(2):1315–1368, 2023). |
| spellingShingle | Berestycki, J Kim, Y Zeitouni, O Mallein, B Lubetzky, E The extremal point process of branching brownian motion in Rd |
| title | The extremal point process of branching brownian motion in Rd |
| title_full | The extremal point process of branching brownian motion in Rd |
| title_fullStr | The extremal point process of branching brownian motion in Rd |
| title_full_unstemmed | The extremal point process of branching brownian motion in Rd |
| title_short | The extremal point process of branching brownian motion in Rd |
| title_sort | extremal point process of branching brownian motion in rd |
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