Sisyphus random walk

TL;DR

This paper studies a special type of random walk with resets to the starting point, analyzing its statistical properties, equilibrium states, and extreme value statistics, with potential applications in physical systems.

Contribution

It introduces and analyzes a specific random walk with restarts, deriving formulas for transition probabilities, equilibrium states, and extreme statistics, and discusses possible generalizations.

Findings

Derived formulas for transition probabilities and equilibrium states.

Analyzed first-passage times and extreme value statistics.

Discussed potential generalizations for physical systems.

Abstract

In this paper we consider a particular version of the random walk with restarts: random reset events which bring suddenly the system to the starting value. We analyze its relevant statistical properties like the transition probability and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks and other extreme statistics are also derived: we consider counting problems associated naturally to the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.