Slowdown in branching Brownian motion with inhomogeneous variance

We consider the distribution of the maximum MT of branching Brownian motion with time-inhomogeneous variance of the form σ(t/T ), where σ(·) is a strictly decreasing function. This corresponds to the study of the time-inhomogeneous Fisher–KolmogorovPetrovskii-Piskunov (FKPP) equation Ft(x, t) = σ (1− t/T )Fxx(x, t)/2 + g(F (x, t)), for appropriate nonlinearities g(·). Fang and Zeitouni (2012) showed that MT − vσT is negative of order T−1/3, where vσ = ∫ 1 0 σ(s)ds. In this paper, we show the existence of a function mT , such that MT − mT converges in law, as T → ∞. Furthermore, mT = vσT −wσT −σ(1) log T +O(1) with wσ = 2α1 ∫ 1 0 σ(s)1/3|σ′(s)|2/3 ds. Here, −α1 = −2.33811... is the largest zero of the Airy function Ai. The proof uses a mixture of probabilistic and analytic arguments.