Waiting times for particles in a branching Brownian motion to reach the rightmost position

We consider a branching Brownian motion (BBM) on the real line R, which evolves as follows. Starting at time t = 0, one particle located at 0, called the root, moves like a standard Brownian motion until an independent exponentially distributed time with parameter 1. At this time it splits into two particles, which, relative to their birth time and position, behave like independent copies of their parent, thus moving like Brownian motions and branching at rate 1 into two copies of themselves. Let N (t) denote the set of all particles alive at time t and let N(t) := #N (t). For any v ∈ N (t) let Xv(t) be the position of v at time t; and for any s < t, let Xv(s) be the position of the unique ancestor of v that was alive at time s. We define R(t) := max u∈N (t) Xu(t) and L(t) := min u∈N (t) Xu(t),