Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks

TL;DR

This paper applies the Real Space Renormalization Group method to analyze extreme value statistics for various stochastic processes, including Brownian motions, Bessel processes, and continuous time random walks, providing new insights and extending known results.

Contribution

It introduces a unified RSRG approach to derive extreme value distributions for multiple stochastic processes, including non-standard fixed points like CTRW.

Findings

Recovered standard extreme value results for Brownian motions.

Extended analysis to Bessel processes and CTRW.

Identified RSRG fixed points for different processes.

Abstract

We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.