论文标题
帕托托环境中关键的加尔顿·瓦特森树的抛物线安德森模型
Parabolic Anderson model on critical Galton-Watson trees in a Pareto environment
论文作者
论文摘要
抛物线安德森模型是热方程，具有一些额外的空间随机性。在本文中，我们考虑使用I.I.D.的抛物线托架模型。占地的帕累托潜力是生存的条件。我们证明，时间$ t $的解决方案集中在一个概率很高的单个站点上，并且在两个站点上肯定是$ t \ to \ infty $。此外，我们确定了本地化站点和总质量的渐近学，并表明解决方案$ u（t，v）$在顶点$ v $可以通过$ v $的某些功能可以很好地x.在$ \ mathbb {z}^d $上与早期结果的主要区别在于，我们必须在树内结合变量顶点度的效果，并确切地使程度的角色。
The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier results on $\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.
