  Persistence, defined as the probability that a fluctuating signal has not
reached a threshold up to a given observation time, plays a crucial role in the
theory of random processes. It quantifies the kinetics of processes as varied
as phase ordering, reaction diffusion or interface relaxation dynamics. The
fact that persistence can decay algebraically with time with non trivial
exponents has triggered a number of experimental and theoretical studies.
However, general analytical methods to calculate persistence exponents cannot
be applied to the ubiquitous case of non-Markovian systems relaxing transiently
after an imposed initial perturbation. Here, we introduce a theoretical
framework that enables the non perturbative determination of persistence
exponents of $d$-dimensional Gaussian non-Markovian processes with general non
stationary dynamics relaxing to a steady state after an initial perturbation.
Two prototypical classes of situations are analyzed: either the system is
subjected to a temperature quench at initial time, or its past trajectory is
assumed to have been observed and thus known. Altogether, our results reveal
and quantify, on the basis of Gaussian processes, the deep impact of initial
perturbations on first-passage statistics of non-Markovian processes. Our
theory covers the case of spatial dimension higher than one, opening the way to
characterize non-trivial reaction kinetics for complex systems with
non-equilibrium initial conditions.
