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The KPP traveling wave in the half-plane

Vortrag von Dr. Bastien Mallein
Datum: 12.06.23   Zeit: 11.00 - 12.00   Raum: Y27H28

H. Berestycki and Cole G. (2022) proved that the F-KPP reaction-diffusion equation $\partial_t u = \frac{1}{2} \Delta u + u(1-u)$ in the half-place with Dirichlet boundary conditions admit traveling waves for all speed $c \geq \sqrt{2}$.

Using the duality between this PDE and the branching Brownian motion in the half-plane with absorption at the boundary, we prove that the minimal speed traveling wave is in fact unique (up to shift). Moreover, we give a probabilistic representation of this traveling wave $\Phi$ in terms of the Laplace transform of a certain "derivative martingale" of this branching Brownian motion.

We use this probabilistic representation to describe the asymptotic behavior of $\Phi$ away from the boundary of the domain, proving that

\[ \lim_{y \to \infty} \Phi\left(x + \frac{1}{\sqrt{2}} \log y, y\right) = w(x) \] where $w$ is the usual one-dimensional traveling wave.

This talk is based on joint work with Julien Berestycki, Cole Graham and Yujin H. Kim.