  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.
