First passage properties of a generalized Polya urn

A generalized two-component Polya urn process, parameterized by a variable alpha, is studied in terms of the likelihood that due to fluctuations the initially smaller population in a scenario of competing population growth eventually becomes the larger, or is the larger after a certain passage of time. By casting the problem as an inhomogeneous directed random walk we quantify this role-reversal phenomenon through the first passage probability that equality in size is first reached at a given time, and the related exit probability that equality in size is reached no later than a given time. Using an embedding technique, exact results are obtained which complement existing results and provide new insights into behavioural changes (akin to phase transitions) which occur at defined values of alpha.