A scaling limit theorem for a class of superdiffusions. (English) Zbl 1014.60080

Engländer, János; Turaev, Dmitry

Ann. Probab. 30, No. 2, 683-722 (2002).

The authors study the superdiffusion \(X_t(dx)\) corresponding to the evolution equation \(u_t = Lu + \beta(x) u - f(x,u)\), where \(\beta\) represents the linear growth rate and \(f\) is a function with some canonical representation. Under appropriate spectral theoretical assumptions, it is proved that the limit \(X = \lim_{t \to\infty} e^{-\lambda_ct} X_t(dx)\) exists in the vague topology, where \(\lambda_c\) is the general principal eigenvalue of \(L+\beta\) on \(R^d\) and it is assumed to be positive and finite, which completes an earlier result of R. G. Pinsky [Ann. Probab. 24, No. 1, 237-267 (1996; Zbl 0854.60087)] on the expectation of the rescaled process. It is also proved that this limiting random measure \(X\) is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator \(L+\beta\). When \(\beta\) is bounded above, \(X\) is finite measure-valued. In this case, under some additional assumption, the convergence is established in the weak topology. As a particular case, if \(L\) corresponds to a positive recurrent diffusion and \(\beta\) is a positive constant, then \(X = \lim_{t \to\infty} e^{-\beta t} X_t(dx)\) exists and is a nonnegative nondegenerate random multiple of the invariant measure of \(L\). Taking \(L = \Delta/2\) on \(R\) and replacing \(\beta\) by \(\delta_0\), the model is known as the super Brownian motion with a single point sauce. In this case, it is shown that a similar result holds with \(\lambda_c\) replaced by \(1/2\) and with the deterministic measure \(e^{|x|}dx\), giving an affirmative answer to a problem of J. Engländer and K. Fleischmann [Stochastic Processes Appl. 88, 37-58 (2002)]. The proofs of the above results are based on some new results on invariant curves of strongly continuous nonlinear semigroups.

Reviewer: Zenghu Li (Beijing)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J60	Diffusion processes
60G57	Random measures

Keywords:

measure-valued process; scaling limit; invariant curve

Citations:

Zbl 0854.60087

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