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