Large Deviations for the Rightmost Position in a Branching Brownian Motion

Abstract

Dedicated to Professor Valentin Konakov on the occasion of his 70th birthday Summary. We study the lower deviation probability of the position of the rightmost particle in a branching Brownian motion and obtain its large deviation function.

Large deviations for the right-most position of a last progeny modified branching random walk

“…Also, parts (iii) and (iv) of Theorem 1.2 calculated by Gantert and Höfelsauer [7] have been simplified here. Notice here that the rate function in Theorem 1.2 is similar to that of the Branching Brownian Motion (BBM) calculated by Derrida and Shi [6]. As a result, we also see similarities in the figures in Section 2 and those in Derrida and Shi [6].…”

Lower deviation and moderate deviation probabilities for maximum of a branching random walk

“…For maximum of branching Brownian motion, Chauvin and Rouault [13] first studied the large deviation probability. Recently, Derrida and Shi [16,17,18] considered both the large deviation and lower deviation. They established precise estimations.…”

“…Motivated by [25], [23] and [17], the goal of this article is to study moderate deviation P(M n ≤ x * n − 3 2θ * log n − n ) with n = O(n). As we already mentioned, [25] first considered this problem with n = o(log n); see Remarks 1.2 and 1.5 below for more details.…”

Slower deviations of the branching Brownian motion and of branching random walks

“…Appendix A.1. Derivation of (15) For the branching Brownian motion, starting from (4), we know that F (z) → 1 as z → ∞ and F (z) → Be ( √ 2−1)z as z → ∞ (see (5)). Therefore for any 0 < η < 2(…”

“…In the present work we estimate the long time asymptotics of u(x, t) for x t = c < 2. This large deviation function exhibits a phase transition [15] at some negative velocity…”