Random walks in noninteger dimension

One can define a random walk on a hypercubic lattice in a space of integerdimension $D$. For such a process formulas can be derived that express theprobability of certain events, such as the chance of returning to the originafter a given number of time steps. These formulas are physically meaningfulfor integer values of $D$. However, these formulas are unacceptable asprobabilities when continued to noninteger $D$ because they give values thatcan be greater than $1$ or less than $0$. In this paper we propose a randomwalk which gives acceptable probabilities for all real values of $D$. This$D$-dimensional random walk is defined on a rotationally-symmetric geometryconsisting of concentric spheres. We give the exact result for the probabilityof returning to the origin for all values of $D$ in terms of the Riemann zetafunction. This result has a number-theoretic interpretation.Comment: 25 pages, 5 figures included, 2 figures on request, plain TE