Wave Propagation for Reaction-Diffusion Equations on Infinite Random Trees


The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees dβ†’ and the random branch lengths β„“β†’ of the tree 𝕋dβ†’,β„“β†’. This speed is slower than that of the same equation on the real line R, and we estimate this slow down in terms of dβ†’ and β„“β†’. The key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [31], with skewness and interface sets that encode the metric structure (dβ†’ , β„“β†’) of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of 2 x 2 random matrices parametrized by dβ†’ and β„“β†’ and for hitting times of a random walk in random environment.


Mathematics and Statistics

Research Center/Lab(s)

Intelligent Systems Center


National Science Foundation, Grant 1804492

International Standard Serial Number (ISSN)

0010-3616; 1432-0916

Document Type

Article - Journal

Document Version


File Type





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Publication Date

01 May 2021