Persistence with Partial Survival

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 $ϵ=1-p$, for a general Gaussian process with finite density of zero crossings.