Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift $-\rho$, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten showed that this process dies out with probability one if and only if $\rho \geq \sqrt{2}$. In this talk I will focus on the subcritical case when $\rho > \sqrt{2}$ and consider two questions:

Firstly, can we obtain Yaglom type results? That is if we condition the process to be alive at large time $t$ what do we see? And if there is some sort of limit distribution, how does it behave when $\rho \searrow \sqrt 2$?

Second, we will study a question related to the so-called consistent maximal displacement: what is the inital position that a partcile must start from so that the process can survive until a large time $t$? We find that there is a phase transition, with the behavior changing when $t$ is of the order $\eps^{-3/2}$.

This is based on joint works with Jiaqi Liu, Bastien Mallein, and Jason Schweinsberg.