It is shown that the two descriptions of the ages of alleles corresponding to the two formulations of the stationary infinitely-many-neutral-alleles diffusion model discussed by Ethier (1990a) are equivalent.

We consider two independent Dawson-Watanabe super-Brownian motions, Y1 and Y2. These processes are diffusions taking values in the space of finite measures on ℝd. We show that if d ≤ 5 then with positive probability there exist times t such that the closed supports of intersect; whereas if d > 5 then no such intersections occur. For the case d ≤ 5, we construct a continuous, non-decreasing measure–valued process L(Y1, Y2), the collision local time, such that the measure defined by , is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition of L(Y1, Y2). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions with point interactions. In the course of our proofs we obtain smoothness results for the random measures that are uniform in t. These theorems use a nonstandard description of Yi and are of independent interest.

We discuss two formulations of the infinitely-many-neutral-alleles diffusion model that can be used to study the ages of alleles. The first one, which was introduced elsewhere, assumes values in the set of probability distributions on the set of alleles, and the ages of the alleles can be inferred from its sample paths. We illustrate this approach by proving a result of Watterson and Guess regarding the probability that the most frequent allele is oldest. The second diffusion model, which is new, assumes values in the set of probability distributions on the set of pairs (x, a), where x is an allele and a is its age. We illustrate this second approach by proving an extension of the Ewens sampling formula to age-ordered samples due to Donnelly and Tavaré.