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 \(\vec {d}\) and the random branch lengths \(\vec {\ell }\) of the tree \(\mathbb {T}_{\vec {d}, \vec {\ell }}\). This speed is slower than that of the same equation on the real line \(\mathbb {R}\), and we estimate this slow down in terms of \(\vec {d}\) and \(\vec {\ell }\). 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 \((\vec {d}, \vec {\ell })\) 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 \(2\times 2\) random matrices parametrized by \(\vec {d}\) and \(\vec {\ell }\) and for hitting times of a random walk in random environment.

考虑了一类无限随机度量树上FKPP型反应扩散方程的渐近波速度。我们表明，只要反应速率足够大，就会出现行波前。波前的传播速度可以通过一个变分公式来量化，该变分公式涉及树的随机分支度\（\ vec {d} \）和树的随机分支长度\（\ vec {\ ell} \）\（\ mathbb {T} _ {\ vec {d}，\ vec {\ ell}} \）。该速度比实线\（\ mathbb {R} \）上相同方程的速度慢，并且我们根据\（\ vec {d} \）和\（\ vec {\ ell } \）。关键思想是将树上的布朗运动沿波传播方向投影到一维轴上。投影过程是由Ramirez [31]引入的多斜布朗运动，其斜度和接口集对树的度量结构\（（\ vec {d}，\ vec {\ ell}）\）进行编码。结合基于Feynman-Kac公式的解析论证，该思想将我们对波前传播的分析与具有随机偏斜度和随机界面集的多偏斜布朗运动的大偏差原理（LDP）相联系。我们的LDP分析涉及对\（\ vec {d} \）和\（\ vec {\ ell} \）参数化的\（2 \ times 2 \）随机矩阵的无限乘积的精细估计。 以及在随机环境中的随机行走击中时间。