Finite time and asymptotic behaviour of the maximal excursion of a random walk

We evaluate the limit distribution of the maximal excursion of a random wal k in any dimension for homogeneous environments and for self-similar suppor ts under the assumption of spherical symmetry. This distribution is obtaine d in closed form and is an approximation of the exact distribution comparab le to that obtained by real space renormalization methods. We then focus on the early time behaviour of this quantity. The instantaneous diffusion exp onent v(n) exhibits a systematic overshooting of the long-time exponent. Ex act results are obtained in one dimension up to third order in n(-1/2). In two dimensions, on a regular lattice and on the Sierpinski gasket we find n umerically that the analytic scaling v(n) similar or equal to v + An(-v) ho lds.