A scaling limit theorem for a class of superdiffusions

• 成果类型: Article
• 署名作者: Engländer, L; Turaev, D
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 683-722
• 关键词: extinction properties; support

摘要

Consider the or-finite measure-valued diffusion corresponding to the evolution equation u(t) = Lu + beta(x)u - f(x, u), where f(x, u) = alpha(x)u(2) + integral(0)(infinity) (e(-ku) - 1 + ku)n (x, dk) and n is a smooth kernel satisfying an integrability condition. We assume that beta, alpha is an element of C-eta (R-d) with eta is an element of (0, 1), and alpha > 0. Under appropriate spectral theoretical assumptions we prove the existence of the random measure lim(tup arrowinfinity)(e-lambdact)X(t)(dx) (with respect to the vague topology), where lambda(c) is the generalized principal eigenvalue of L + beta 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 + beta. When beta is bounded from above, X is finite measure-valued. In this case, under an additional assumption on L + beta 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 beta is a positive constant, then lim(tup arrowinfinity) e(-betat) X-t (dx) exists and equals a nonnegative nondegenerate random multiple of the invariant measure for Y. Taking L = (1)/(2) Delta on R and replacing beta by delta(0) (super-Brownian motion with I a single point source), we prove a similar result with lambda(c) replaced by (1)/(2) and 2 with the deterministic measure e(-\x\)dx, giving an answer in the affirmative to a problem proposed by Englander 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.