Analytical results for generalized persistence properties of smooth processes

We present a general scheme to calculate within the independent interval ap proximation generalized (level-dependent) persistence properties for proces ses having a finite density of zero crossings. Our results are especially r elevant for the diffusion equation evolving from random initial conditions- one of the simplest coarsening systems. Exact results:are obtained in certa in limits, and rely on a new method to deal with constrained multiplicative processes. An excellent agreement of our analytical predictions with direc t numerical simulations of the diffusion equation is found.