Fig. 2: Persistence for fBM conditioned on the past trajectory (type II problem).
From: Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks
a Definition of the problem: assume that a particular trajectory is observed for t < 0 (black curve). The blue line is one realization of trajectory for t > 0, the dashed red line represents the average trajectory given that the past trajectory is observed. The time t* is the first crossing time to this predicted average trajectory, and the persistence exponent characterizes the probability that t* > t, for large t. b Comparison between values of θ for the fBM conditioned on the past trajectory, obtained in simulations (symbols), our theoretical approach (black line) and perturbation expansion (dashed red line). We also indicate the value θ = 1 − H for non-conditioned fBM.
