A scaling limit theorem for a class of superdiffusions

Consider the σ-finite measure-valued diffusion corresponding to the evolution equation u_t = Lu + β(x)u - f(x, u), where f(x,u) = α(x)u² + ∫₀^∞ (e^-ku - 1 + ku)n(x,dk) and n is a smooth kernel satisfying an integrability condition. We assume that β, α ∈ C^η(ℝ^d) with η ∈ (0, 1], and α > 0. Under appropriate spectral theoretical assumptions we prove the existence of the random measure lim_t↑∞ e^-λctX_t(dx) (with respect to the vague topology), where λ_c is the generalized principal eigenvalue of L + β on ℝ^d and it is assumed to be finite and positive, completing a result of Pinsky on the expectation of the rescaled process. Moreover, we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator L + β. When β is bounded from above, X is finite measure-valued. In this case, under an additional assumption on L + β, we can actually prove the existence of the previous limit with respect to the weak topology. As a particular case, we show that if L corresponds to a positive recurrent diffusion Y and β is a positive constant, then lim_t↑∞ e^-βt X_t(dx) exists and equals a nonnegative nondegenerate random multiple of the invariant measure for Y. Taking L = 1/2 δ on ℝ and replacing β by δ₀ (super-Brownian motion with a single point source), we prove a similar result with λ_c replaced by 1/2 and with the deterministic measure e^-|x|dx, giving an answer in the affirmative to a problem proposed by Engländer and Fleischmann [Stochastic Process. Appl. 88 (2000) 37-58]. The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.