The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift

Preprint

Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift −ρ, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten [14] showed that this process dies out with probability one if and only if ρ ≥ √2. We show that in the subcritical case when ρ > √2, the law of the process conditioned on survival until time t converges as t → ∞ to a quasi-stationary distribution, which we call the Yaglom limit. We give a construction of this quasi-stationary distribution. We also study the asymptotic behavior as ρ ↓ √2 of this quasi-stationary distribution. We show that the logarithm of the number of particles and the location of the highest particle are of order ε−1/3, and we obtain a limit result for the empirical distribution of the particle locations.

Authors: Berestycki, J; Liu, J; Mallein, B; Schweinsberg, J

• Publication status: Published
• Peer review status: Not peer reviewed
• Preprint server: arXiv
• Publication date: 2024-09-16
• DOI: 10.48550/arxiv.2409.08789
• Language: English
• Keywords: Branching Brownian motion; Yaglom limit; quasi-stationary distribution