Spatial Extent of Branching Brownian Motion

We study the one dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost ($X_{\max}\geq 0$) and leftmost ($X_{\min} \leq 0$) visited sites up to time $t$. At each time step the existing particles in the system either diffuse (with diffusion constant $D$), die (with rate $a$) or split into two particles (with rate $b$). We focus on the regime $b \leq a$ where these two extreme values $X_{\max}$ and $X_{\min}$ are strongly correlated. We show that at large time $t$, the joint probability distribution function (PDF) of the two extreme points becomes stationary $P(X,Y,t \to \infty) \to p(X,Y)$. Our exact results for $p(X,Y)$ demonstrate that the correlation between $X_{\max}$ and $X_{\min}$ is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF $p(ζ)$ of the (dimensionless) span $ζ= {(X_{\max} - X_{\min})}/{\sqrt{D/b}}$, which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior ${p}(ζ) \sim \frac{1}{2} \left(1 + Δ\right) ζ$ for small spans, with $Δ= \left(\frac{a}{b} -1\right)$. In the critical case ($Δ= 0$) this distribution has a non-trivial power law tail ${p}(ζ) \sim 8 π\sqrt{3} /ζ^3$ for large spans. On the other hand, in the subcritical case ($Δ> 0$), we show that the span distribution decays exponentially as ${p}(ζ) \sim (A^2/2) ζ\exp \left(- \sqrtΔ~ζ\right)$ for large spans, where $A$ is a non-trivial function of $Δ$ which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between $X_{\max}$ and $X_{\min}$. Finally we verify our results via direct Monte Carlo simulations.

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