Unified approach to reset processes and application to coupling between process and reset

Processes under reset, where realizations are interrupted according to some stochastic rule and restarted from the initial state, find broad application in modeling physical, chemical, and biological phenomena and in designing search strategies. While a wealth of theoretical results has been recently obtained, current derivations tend to focus on specific processes, obscuring the general principles and preventing broad applicability. 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 general solutions, and determine the conditions for existence and equality of stationary values with and without reset. For intermittent (i.e., broadly distributed) reset times, we derive exact asymptotic expressions for observables that vary asymptotically as a power of time. We illustrate the general approach with results for occupation densities and moments of subdiffusive processes. We focus on subdiffusion-decay processes with microscopic dependence between transport and decay, where the probability of a random walker to be removed and subsequently restarted depends on the local transit times. In contrast to the uncoupled case, restarting the particle upon decay does not produce a probability current associated with restart equal to the decay rate, but instead drastically alters the time dependence of the decay rate and the resulting current due to memory effects associated with ageing. Our framework shows that such effects are independent of the specific microscopic details, uncovering the general impact of restart on occupation densities, spatial moments, and other quantities.