  In this paper we address the one-dimensional problem of stochastic renewal in
different damping environments. An ensemble of particles with some specified
initial distribution in phase space are allowed to evolve stochastically till a
certain instant of time (say,$tau$), when a restoring force is applied to bring
them back to some point in configuration space. The physical quantities of
interest that have been studied are the Survival Probability and the First
Passage distribution for return to the specified target point. We observe
nontrivial dependence of these quantities on $tau$ as well as on the width of
the initial distribution, which has been taken to be Gaussian in position and
velocity.
