Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks

Abstract

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 1,2 . 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 3-7 and theoretical 8-18 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.The persistence S(t) is the probability that a random process x(t) has not reached a threshold up to time t 1,2 . This quantity is a natural tool in non equilibrium statistical physics to probe the history of various systems undergoing phase ordering 10,11,19 or reaction diffusion dynamics 2 , or to quantify the efficiency of target search problems [20][21][22][23][24][25][26][27][28][29] .It has been recognized that the long time decay of persistence is often algebraic, S(t) ∼ t −θ , where the persistence exponent θ is non trivial as soon as the process is non-Markovian (i.e. displays memory effects).As a matter of fact, even for seemingly simple Gaussian dynamics where all correlation functions are known, θ is generally non-trivial and not known in closed form. This has triggered an intense theoretical activity for its determination. Existing approaches to quantify persistence exponents of Gaussian processes can be classified according to the nature, stationary or not, of the increments x(t + τ ) − x(t). If these increments are stationary at all times, meaning that their statistics do not depend on the observation time t (such as in the case of the fractional Brownian motion), θ is exactly known 17,18,30 58 . In the opposite case where the increments always depend on the observation