  We present a unified approach to those observables of stochastic processes
under reset that take the form of averages of functionals depending on the most
recent renewal period. We derive solutions for the observables, and determine
the conditions for existence and equality of their stationary values with and
without reset. For intermittent reset times, we derive exact asymptotic
expressions for observables that vary asymptotically as a power of time. We
illustrate the general approach with general and particular results for the
power spectral density, and moments of subdiffusive processes. We focus on
coupling of the process and reset via a diffusion-decay process with
microscopic dependence between transport and decay. In contrast to the
uncoupled case, we find that restarting the particle upon decay does not
produce a probability current equal to the decay rate, but instead drastically
alters the time dependence of the decay rate and the resulting current.
