Persistence with Partial Survival

Abstract

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