Striking universalities in stochastic resetting processes

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).