The continuum P\'olya-like random walk

Abstract

The P\'olya urn scheme is a discrete-time process concerning the addition and removal of colored balls. There is a known embedding of it in continuous-time, called the P\'olya process. We deal with a generalization of this stochastic model, where the initial values and the entries of the transition matrix (corresponding to additions or removals) are not necessarily fixed integer values as in the standard P\'olya process. In one of the scenarios, we even allow the entries of the matrix to be random variables. As a result, we no longer have a combinatorial model of "balls in an urn," but a broader interpretation as a random walk in a possibly high number of dimensions. In this paper, we study several parametric classes of these generalized continuum P\'olya-like random walks.