Article,
Average trajectory of returning walks
Phys. Rev. E, (2003)
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, 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.
