Persistence with Partial Survival

TL;DR

This paper introduces the concept of partial survival in stochastic processes, analyzing how the persistence exponent varies with the parameter p, and provides exact and approximate calculations for specific models.

Contribution

It defines partial survival in persistence, computes the exponent exactly for a coarsening model, and develops a series expansion for Gaussian processes.

Findings

Persistence exponent varies continuously with p for smooth processes

Exact calculation of (p) for a 1D coarsening model

Series expansion for (p) in Gaussian processes

Abstract

We introduce a parameter $p$, called partial survival, in the persistence of stochastic processes and show that for smooth processes the persistence exponent $\theta(p)$ changes continuously with $p$, $\theta(0)$ being the usual persistence exponent. We compute $\theta(p)$ exactly for a one-dimensional deterministic coarsening model, and approximately for the diffusion equation. Finally we develop an exact, systematic series expansion for $\theta(p)$, in powers of $\epsilon=1-p$, for a general Gaussian process with finite density of zero crossings.