Random walks and branching random walks: old and new perspectives - Lecture 14

This course will focus on two well-studied models of modern probability: simple symmetric and branching random walks in ℤd. The focus will be on the study of their trace in the regime that this is a small subset of the ambient space.
We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

A branching random walk (or tree indexed random walk) in ℤd is a non-Markovian process whose time index is a random tree. The random tree is either a critical Galton Watson tree or a critical Galton Watson tree conditioned to survive. Each edge of the tree is assigned an independent simple random walk in ℤd increment and the location of every vertex is given by summing all the increments along the geodesic from the root to that vertex. When $d\geq 5$, the branching random walk is transient and we will mainly focus on this regime. We will introduce the notion of branching capacity and show how it appears naturally as a suitably rescaled limit of hitting probabilities of sets. We will then use it to study covering problems analogously to the random walk case.