Annihilating branching Brownian motion

TL;DR

This work analyzes annihilating branching Brownian motion (ABBM) on the real line with multiple colors that annihilate upon contact. It develops a robust martingale framework (additive and derivative) and innovative couplings (conservative and enhanced) to establish positive probability of coexistence for two or more colors and to characterize the limiting speed of the interface separating colors. The authors prove the interface speed exists on the coexistence event, has no atoms in (−√2,√2), and is almost surely linear with a random slope, while endpoints ±√2 are ruled out via derivative martingale analysis. They also extend the coexistence and speed analysis to multi-type ABBM and discuss generalizations to other annihilating spatial branching processes and higher dimensions, highlighting open problems and directions for future work.

Abstract

We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.

Figures (4)

• Figure 1: ABBM started with particles at $\pm1$. Each horizontal slice is a configuration of particles at a moment in time.
• Figure 2: An ABBM interface.
• Figure 3: The number of particles $n_i$ at positions $a_i$ are chosen along a bell-shaped curve in such a way that, with positive probability, each type $i$ survives in the vicinity of $\lambda_i t$. In this figure, $k=7$ and $\delta=1$.
• Figure 4: Simulations of annihilating branching random walk on $\mathbb Z^2$, started with up 2 or 4 particles of each type. The top two are the same process at times 50 and 256. The bottom two are distinct runs at time 256.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Lemma 3
• Proposition 4
• Theorem 5: Madaule Mad16
• Theorem 6: Arguin et al. ABK13
• Proposition 7
• Lemma 8
• proof
• proof : Proof of Proposition \ref{['prop:mart_con']}