Average trajectory of returning walks

TL;DR

This paper analyzes the average shape of excursions in one-dimensional stochastic processes, revealing universal semicircular forms for uncorrelated walks and complex shapes under correlated noise.

Contribution

It introduces a scaling form for the average trajectory shape during excursions and explores universality and deviations across different stochastic processes.

Findings

Uncorrelated random walks have a semicircular average shape.

Biased walks and Levy flights mostly share this universality, except for some Levy flights.

Correlations and damping alter the shape, leading to asymmetries or flat excursions.

Abstract

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, independently 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 non trivial asymmetric shapes, which are studied numerically.