Reinforced Brownian Motion on the Half-Line

We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half line. A reinforced 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 $N$^{th}$ 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 of the reinforcement.