From first-passage times of random walks in confinement to geometry-controlled kinetics

We present a general theory which allows one to accurately evaluate the mean first-passage time (FPT) for regular random walks in bounded domains, and its extensions to related first-passage observables such as splitting probabilities and occupation times. It is showed that this analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source–target distance in the case of general scale invariant processes. This analysis is applicable to a broad range of stochastic processes characterized by scale-invariance properties. The full distribution of the FPT can be obtained using similar tools, and displays universal features. This allows to quantify the fluctuations of the FPT in confinement, and to reveal the key role that can be played by the starting position of the random walker. Applications to reaction kinetics in confinement are discussed.