  We report numerical and analytic results for the spatial survival probability
for fluctuating one-dimensional interfaces with Edwards-Wilkinson or
Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are
obtained from analysis of steady-state profiles generated by integrating a
spatially discretized form of the Edwards-Wilkinson equation to long times. We
show that the survival probability exhibits scaling behavior in its dependence
on the system size and the `sampling interval' used in the measurement for both
`steady-state' and `finite' initial conditions. Analytic results for the
scaling functions are obtained from a path-integral treatment of a formulation
of the problem in terms of one-dimensional Brownian motion. A `deterministic
approximation' is used to obtain closed-form expressions for survival
probabilities from the formally exact analytic treatment. The resulting
approximate analytic results provide a fairly good description of the numerical
data.
