A strong law of large numbers for super-stable processes

Let ℓ be Lebesgue measure and X=(Xt,t≥0;Pμ) be a supercritical, super-stable process corresponding to the operator −(−Δ)α/2u+βu−ηu2 on Rd with constants β,η>0 and α∈(0,2]. Put Wˆt(θ)=e(|θ|α−β)tXt(e−iθ⋅), which for each smallθ is an a.s. convergent complex-valued martingale with limit Wˆ(θ) say. We establish for any starting finite measure μ satisfying ∫Rd|x|μ(dx)<∞ that td/αXteβt→cαWˆ(0)ℓPμ-a.s. in a topology, termed the shallow topology, strictly stronger than the vague topology yet weaker than the weak topology, where cα>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.