A scaling limit theorem for a class of superdiffusions
- Engländer, János
- Turaev, Dmitry
2010 Mathematics Subject Classification
- 60J80 60J60 60G57 37L25 35B40 35K55
- measure-valued process, superprocess, super-Brownian motion, scaling limit, single point source, invariant curve, recurrent diffusion, lambda-lemma, elliptic equation, parabolic PDE
ƒ (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 ∈ Cη(ℝd) with η∈(0,1], and α > 0.
Under appropriate spectral theoretical assumptions we prove the existence of the random measure
lim e-λct Xt (dx)
(with respect to the vague topology), where λc is the principal eigenvalue of L + β on ℝd 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 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 e-βt Xt (dx)
exists and equals to a nonnegative nondegenerate random multiple of the invariant measure for Y.
Taking L = ½ Δ on ℝ and replacing β by δ0 (super-Brownian motion with a single point source), we prove a similar result with λc replaced by ½ and with the deterministic measure e-IxIdx. The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.