Striking universalities in stochastic resetting processes

TL;DR

This paper uncovers universal probability laws in stochastic resetting processes, showing that certain event probabilities are independent of the specific process, observables, and resetting distribution, revealing deep underlying symmetries.

Contribution

The study demonstrates super-universal probabilities in stochastic resetting, independent of process details, observables, and resetting distributions, under mild assumptions.

Findings

Probabilities of certain events are process-independent.

Universality holds for various observables and resetting distributions.

Some universality results require mild symmetry assumptions.

Abstract

Given a random process $x(\tau)$ which undergoes stochastic resetting at a constant rate $r$ to a position drawn from a distribution ${\cal P}(x)$, we consider a sequence of dynamical observables $A_1, \dots, A_n$ associated to the intervals between resetting events. We calculate exactly the probabilities of various events related to this sequence: that the last element is larger than all previous ones, that the sequence is monotonically increasing, etc. Remarkably, we find that these probabilities are ``super-universal'', i.e., that they are independent of the particular process $x(\tau)$, the observables $A_k$'s in question and also the resetting distribution ${\cal P}(x)$. For some of the events in question, the universality is valid provided certain mild assumptions on the process and observables hold (e.g., mirror symmetry).