A random walk with partial reflection at its extrema

We attempt an in-depth study of a so-called reinforced random process which behaves like a simple (fair) random walk except when at its previous extreme values. We say this process has partially reflecting extrema because at its maximum or minimum, it will more likely be reflected back into previously visited states. (Alternatively, it may be more likely to be attracted into unvisited areas, in which case the process has partially attractive extrema.) Other investigations of general reinforced processes tell us some properties of this walk; by exploring this random walk specifically, we discover some stronger results about the process\u27 long-term behavior. In particular, we examine the limiting behavior of the process with respect to a certain stopping time, and we explore the analog of the classical gambler\u27s ruin question