A strong law of large numbers for super-stable processes

Let ℓ be Lebesgue measure and X = ( X t , t ≥ 0 ; P μ ) be a supercritical, super-stable process corresponding to the operator − ( − Δ ) α / 2 u + β u − η u 2 on R d with constants β , η > 0 and α ∈ ( 0 , 2 ] . Put W ˆ t ( θ ) = e ( | θ | α − β ) t X t ( e − i θ ⋅ ) , which for each small θ is an a.s. convergent complex-valued martingale with limit W ˆ ( θ ) say. We establish for any starting finite measure μ satisfying ∫ R d | x | μ ( d x ) < ∞ that t d / α X t e β t → c α W ˆ ( 0 ) ℓ P μ -a.s. in a topology, termed the shallow topology, strictly stronger than the vague topology yet weaker than the weak topology, where c α > 0 is a known constant. This result can be thought of as an extension to a class of superprocesses of Watanabe’s strong law of large numbers for branching Markov processes.