Zhenyuan Zhang
Stanford University
Room: M3 3127
Branching processes, extremes, and random environments
Abstract: Branching random walks (BRW) describe populations with movement and reproduction. They arise naturally in models of biological invasion, bacterial growth, and viral evolution, and the mathematical literature mostly focused on asymptotics of the maxima of BRW on ℝ. This talk studies extremes of BRW more broadly through understanding asymptotics of first-passage times on state spaces without a natural order, such as Euclidean spaces ℝᵈ and Hamming cubes {0, 1, …, b − 1}ᵈ. BRW on ℝᵈ is motivated from understanding shortest paths in polymer networks. BRW on the Hamming cube is motivated by RNA sequence evolution with the interaction of mutation, reproduction, and selection. We model this variation in fitness through a space-inhomogeneous random environment. Our model indicates that increasing the mutation rate delays the appearance of dangerous mutants, which is key to antiviral mutagenesis. We discuss how the geometry of the state space shapes the effects of the random environment on extremes. This is based on joint work with Jose Blanchet.