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Dostęp zamkniętyBranching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift
| dc.abstract.en | We study a dyadic branching Brownian motion on the real line with absorption at 0, drift μ∈R and started from a single particle at position x>0. With K(∞) the (possibly infinite) total number of individuals absorbed at 0 over all time, we consider the functions ωs(x):=Ex[sK(∞)] for s≥0. In the regime where μ is large enough so that K(∞)<∞ almost surely and that the process has a positive probability of survival, we show that ωs<∞ if and only if s∈[0,s0] for some s0>1 and we study the properties of these functions. Furthermore, ω(x):=ω0(x)=Px(K(∞)=0) is the cumulative distribution function of the all time minimum of the branching Brownian motion with drift started at 0 without absorption. We give descriptions of the family ωs,s∈[0,s0] through the single pair of functions ω0(x) and ωs0(x), as extremal solutions of the Kolmogorov–Petrovskii–Piskunov (KPP) travelling wave equation on the half-line, through a martingale representation, and as a single explicit series expansion. We also obtain a precise result concerning the tail behaviour of K(∞). In addition, in the regime where K(∞)>0 almost surely, we show that u(x,t):=Px(K(t)=0) suitably centred converges to the KPP critical travelling wave on the whole real line. |
| dc.affiliation | Uniwersytet Warszawski |
| dc.contributor.author | Berestycki, Julien |
| dc.contributor.author | Brunet, Eric |
| dc.contributor.author | Harris, Simon |
| dc.contributor.author | Miłoś, Piotr |
| dc.date.accessioned | 2024-01-24T18:54:53Z |
| dc.date.available | 2024-01-24T18:54:53Z |
| dc.date.issued | 2017 |
| dc.description.finance | Nie dotyczy |
| dc.description.number | 6 |
| dc.description.volume | 273 |
| dc.identifier.doi | 10.1016/J.JFA.2017.06.006 |
| dc.identifier.issn | 0022-1236 |
| dc.identifier.uri | https://repozytorium.uw.edu.pl//handle/item/102510 |
| dc.identifier.weblink | http://www.sciencedirect.com/science/article/pii/S0022123617302239 |
| dc.language | eng |
| dc.relation.ispartof | Journal of Functional Analysis |
| dc.relation.pages | 2107-2143 |
| dc.rights | ClosedAccess |
| dc.sciencecloud | nosend |
| dc.subject.en | branching Brownian motion |
| dc.title | Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift |
| dc.type | JournalArticle |
| dspace.entity.type | Publication |