The continuum Pólya-like random walk

Abstract

The Pólya 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 Pólya process. We introduce 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 Pólya process. In one of our 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 Pólya-like random walks.

Random walks exhibiting anomalous diffusion: elephants, urns and the limits of normality

“…As written, the exchange symmetry (a, A) ⇔ (b, B) is no longer obvious; other equivalent forms can be written down but this one is relatively compact. The result for α = 0 was previously given in (30). It should be possible, in principle, to take the limit α → 0 in (49) to recover (30), although we do not pursue that here.…”

“…Recent studies of the original ERW have benefited from a mapping onto an urn process, which has greatly aided understanding of the limiting behaviour as the number of steps grows large [16]. In other fields, the nature of urn-based random walk models has been explored in parallel [26][27][28][29][30][31][32]. We exploit this here by demonstrating asymptotic equivalence of Model I to a non-linear generalization of the standard two-component Pólya urn process [31], which we call Model II.…”