## First-passage time densities for non-Markovian uncoupled continuous-time random walks and the universality of the Sparre Andersen theorem

Non-Markovian continuous-time random walks on homogeneous space are considered in the uncoupled formulation, that is, the i.i.d. random waiting-times and the i.i.d. random jump-sizes are, at any epoch, independent of each other and also of the current position. Non-Markovianity is set by a power-lawed waiting-time density providing an infinite-mean waiting-time, in analogy with the successful modelling of anomalous diffusion in complex systems. First, the evolution equation of the first-passage time density is derived and later the relation of this last with the same density emerging in the Markovian case is obtained. The two densities are related by an integral formula involving the Mainardi (M-Wright) function. On the basis of this result, it is shown that the analogue of the Sparre Andersen theorem in this non-Markovian setting is universal, namely, independent of the jump-size density, as well as in the Markovian case.

Area: IS13 - Non-local operators in probability (Giacomo Ascione)

Keywords: First-passage time density, survival probability, random walks, Sparre Andersen theorem, Wiener–Hopf integral, fractional evolution equations

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