Super-mouvement brownien avec catalyse

We study a general class of catalytic super-Brownian motions. Informally, such a process Z describes the evolution of a large number of small particles which move according to independent Brownian motions in and branch only when they visit a given set D (the catalyst), which may be of zero Lebesgue measure. We first construct the processus Z as a weak limit of branching particle systems. We then obtain detailed information about the continuity properties of Z Using excursion theory for Brownian motion in , we prove a representation theorem for Z outside the catalyst D. In the special case when D has zero Lebesgue measure, this representation theorem shows that a.s. for every t > 0, the measure Zt is absolutely continuous with respect to Lebesgue measure, and its density solves the heat equation outside D