Reinforced Brownian Motion: A Prototype

We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half-line. A reinfoced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of S_N (the location of the walker at the N^t step) and A_N, the maximum location the walker achieved in N steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its "environment", both parametrized by the amplitude δ of the reinforcement. The walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement, details depending very much on the value of -1<δ < ∞.