An almost sure scaling limit theorem for Dawson–Watanabe superprocesses

AbstractWe establish a scaling limit theorem for a large class of Dawson–Watanabe superprocesses whose underlying spatial motions are symmetric Hunt processes, where the convergence is in the sense of convergence in probability. When the underling process is a symmetric diffusion with Cb1-coefficients or a symmetric Lévy process on Rd whose Lévy exponent Ψ(η) is bounded from below by c|η|α for some c>0 and α∈(0,2) when |η| is large, a stronger almost sure limit theorem is established for the superprocess. Our approach uses the principal eigenvalue and the ground state for some associated Schrödinger operator. The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap