On the shape of planar Brownian paths
We establish a formula describing the shape of the convex hull of sample paths in the case of planar Brownian motion : viz. the average number of edges joining paths’ points separated by a time-lapse \Delta \tau in [ \Delta \tau _1, \Delta \tau_2 ] is equal to 2\log (\Delta \tau_2 / \Delta \tau_1 ), regardless of the total duration T of the motion. The formula exhibits invariance when the time scale is multiplied by any factor.
Apart from its theoretical importance, our result provides new insights regarding the shape of two-dimensional objects modelled by stochastic processes’ sample paths (eg polymer chains) : in particular for a total time (or parameter) duration T, the average number of edges on the convex hull ("cut off’’ to discard edges joining points separated by a time-lapse shorter than some \Delta \tau much smaller than T) will be given by 2 \log (T / \Delta \tau). Thus it will only grow logarithmically, rather than at some higher pace.