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