ALMOST SURE FINITENESS FOR THE TOTAL OCCUPATION TIME OF AN (d,α,β)-superprocess

For 0 < α ≤ 2 and 0 < β ≤ 1 the (d,α,β)-superprocess is the superprocess with symmetric α-stable spatial movement in Rd and spectrally positive (1+β)-stable branching. It is a measurevalued process arising as the high density limit of empirical measure for the following critical branching symmetric α-stable particle system. Independent of the others, each particle is assigned a mass n−1 and it branches at rate γn for some constant γ > 0. The offspring distribution of each particle is determined by the generating function