Average trajectory of returning walks

We compute the average shape of trajectories of some one-dimensional stochastic processes x(t) in the (t,x) plane during an excursion, i.e., between two successive returns to a reference value, finding that it obeys a scaling form. For uncorrelated random walks the average shape is semicircular, independent from the single increments distribution, as long as it is symmetric. Such universality extends to biased random walks and Levy flights, with the exception of a particular class of biased Levy flights. Adding a linear damping term destroys scaling and leads asymptotically to flat excursions. The introduction of short and long ranged noise correlations induces nontrivial asymmetric shapes, which are studied numerically.