Extreme Value Statistics of Jump Processes

Abstract: We investigate extreme value statistics (EVS) of general discrete time and continuous space symmetric jump processes. We first show that for unbounded jump processes, the semi-infinite propagator $G_0(x,n)$, defined as the probability for a particle issued from $0$ to be at position $x$ after $n$ steps whilst staying positive, is the key ingredient needed to derive a variety of joint distributions of extremes and times at which they are reached. Along with exact expressions, we extract novel universal asymptotic behaviors of such quantities. For bounded, semi-infinite jump processes killed upon first crossing of zero, we introduce the \textit{strip probability} $\mu_{0,\underline{x}}(n)$, defined as the probability that a particle issued from 0 remains positive and reaches its maximum $x$ on its $n^{\rm th}$ step exactly. We show that $\mu_{0,\underline{x}}(n)$ is the essential building block to address EVS of semi-infinite jump processes, and obtain exact expressions and universal asymptotic behaviors of various joint distributions.