It has been shown by H. Berestycki and G. Cole (2022) that the F-KPP equation $\partial_t u = \frac{1}{2}\Delta u + u(1-u)$ in the half-plane with Dirichlet boundary conditions admits travelling wave solutions for all speed $c\ge c^*=\sqrt{2}$.
We show that the minimal speed traveling wave $\Phi$ is in fact unique (up to shift) and give a probabilistic representation as the Laplace transform of a certain martingale limit associated to the branching Brownian motion with absorption. This representation allows us to study the asymptotic behaviour of $\Phi$ away from the boundary of the domain, proving that
\begin{equation*}
\lim_{y \to \infty} \Phi\left(x + \tfrac{1}{\sqrt{2}}\log y, y\right) = w(x)
\end{equation*}
where $w$ is the usual one-dimensional critical travelling wave.
We are able to extend our result to the case of the half-space $\mathbb{H}^d =\{x\in R^d : x_1\ge 0\}$. Finally, if time allows, I will also mention some results regarding the convergence towards the critical travelling wave.
This is based on joint work with Cole Graham, Yujin Kim and Bastien Mallein.
