EXCEPTIONAL TIMES FOR THE INSTANTANEOUS PROPAGATION OF SUPERPROCESS

For a Dawson-Watanabe superprocess X on R^d, it is shown by Perkins [Ann. Probab. 18 (1990), pp. 453–491] that if the underlying spatial motion belongs to a particular class of Lévy processes that admit jumps, then for any fixed t>0, the closed support of Xt is the whole space almost surely when conditioned on {X_t≠0}, the so-called “instantaneous propagation” property. In this paper, for the superprocess on R^dwhose spatial motion is the symmetric stable process of index (Formula present), we prove that there exist exceptional times at which the support is compact and nonempty. Moreover, we show that the set of exceptional times is dense with a full Hausdorff dimension. Besides, we prove that near extinction, the support of the superprocess is concentrated arbitrarily close to the extinction point, thus upgrading the corresponding results by Tribe [Ann. Probab. 20 (1992), pp. 286–311] from (Formula present) and d=1 to (Formula present) and d≥1. We further show that the set of such exceptional times also admits a full Hausdorff dimension.