Collision Local Times and Measure-Valued Processes

Barlow, Martin T. ; Evans, Steven N. ; Perkins, Edwin A.

Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 897-938

Résumé

We consider two independent Dawson-Watanabe super-Brownian motions, Y 1 and Y 2. 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(Y 1, Y 2), 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(Y 1, Y 2). 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 Y i and are of independent interest.

Détail

DOI : 10.4153/CJM-1991-050-6

Mots-clés : 60G17, 60G57, 60J55, 60H99, measure-valued diffusion, superprocess, local time, Tanaka formula, additive functional, Hausdorff measure, nonstandard analysis

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