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The first-passage time (FPT), defined as the time a random walker takes toreach a target point in a confining domain, is a key quantity in the theory ofstochastic processes. Its importance comes from its crucial role to quantifythe efficiency of processes as varied as diffusion-limited reactions, targetsearch processes or spreading of diseases. Most methods to determine the FPTproperties in confined domains have been limited to Markovian (memoryless)processes. However, as soon as the random walker interacts with itsenvironment, memory effects can not be neglected. Examples of non Markoviandynamics include single-file diffusion in narrow channels or the motion of atracer particle either attached to a polymeric chain or diffusing in simple orcomplex fluids such as nematics \cite{turiv2013effect}, dense soft colloids orviscoelastic solution. Here, we introduce an analytical approach to calculate,in the limit of a large confining volume, the mean FPT of a Gaussiannon-Markovian random walker to a target point. The non-Markovian features ofthe dynamics are encompassed by determining the statistical properties of thetrajectory of the random walker in the future of the first-passage event, whichare shown to govern the FPT kinetics.This analysis is applicable to a broadrange of stochastic processes, possibly correlated at long-times. Ourtheoretical predictions are confirmed by numerical simulations for severalexamples of non-Markovian processes including the emblematic case of theFractional Brownian Motion in one or higher dimensions. These results show, onthe basis of Gaussian processes, the importance of memory effects infirst-passage statistics of non-Markovian random walkers in confinement.

Mean first-passage times of non-Markovian random walkers in confinement