2023/05/17 by Florian Kreten, Kreten, Florian
Mathematics · #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2305.10228
We construct the traveling wave solutions of an FKPP growth process of two densities of particles, and prove that the critical traveling waves are locally stable in a space where the perturbations can grow exponentially at the back of the wave. The considered reaction-diffusion system was introduced by Hannezo et al. in the context of branching morphogenesis (Cell, 171(1):242-255.e27, 2017): active, branching particles accumulate inactive particles, which do not react. Thus, the system features a continuum of steady state solutions, complicating the analysis. We adopt a result by Faye and Holzer (J.Diff.Eq., 269(9):6559-6601, 2020) for proving the stability of the critical traveling waves, by modifying the semi-group estimates to a space with unbounded weights. The novelty is that we use a Feynman-Kac formula to get an exponential a-priori estimate for the left tail of the PDE, in the regime where the weight is unbounded.