Limit theorem for derivative martingale at criticality w.r.t branching Brownian motion

Abstract

We consider a branching Brownian motion on R in which one particle splits into 1 + X children. There exists a critical value λ ¯ in the sense that λ ¯ is the lowest velocity such that a traveling wave solution to the corresponding Kolmogorov–Petrovskii–Piskunov equation exists. It is also known that the traveling wave solution with velocity λ ¯ is closely connected with the rescaled Laplace transform of the limit of the so-called derivative martingale ∂ W t ( λ ¯ ) . Thus special interest is put on the property of its limit ∂ W ( λ ¯ ) . Kyprianou [Kyprianou, A.E., 2004. Traveling wave solutions to the K–P–P equation: alternatives to Simon Harris’ probability analysis. Ann. Inst. H. Poincaré 40, 53–72.] proved that, ∂ W ( λ ¯ ) > 0 if E X ( log + X ) 2 + δ < + ∞ for some δ > 0 while ∂ W ( λ ¯ ) = 0 if E X ( log + X ) 2 − δ = + ∞ . It is conjectured that ∂ W ( λ ¯ ) is non-degenerate if and only if E X ( log + X ) 2 < + ∞ . The purpose of this article is to prove this conjecture.