Random walks with shrinking steps: First-passage characteristics

We study the mean first-passage time of a one-dimensional random walker with step sizes decaying exponentially in discrete time. That is step sizes go like lambda(n) with lambda <= 1. We also present, for pedagogical purposes, a continuum system with a diffusion constant decaying exponentially in continuous time. Qualitatively both systems are alike in their global properties. However, the discrete case shows very rich mathematical structure, depending on the value of the shrinking parameter, such as self-repetitive and fractal-like structure for the first-passage characteristics. The results we present show that the most important quantitative behavior of the discrete case is that the support of the distribution function evolves in time in a rather complicated way in contrast to the time independent lattice structure of the ordinary random walker. We also show that there are critical values of lambda defined by the equation lambda(K)+2 lambda(P)-2=0 with {K,N}is an element of N where the mean first-passage time undergoes transitions.