A scaling limit theorem for a class of superdiffusions

Abstract

Consider the $\sigma$-finite measure-valued diffusion corresponding to the evolution equation $u_t = Lu + \beta (x) u - f(x,u)$, where

$$f(x,u) = \alpha (x)u^2 + \int_0^{\infty} (e^{-ku} - 1 + ku)n(x,dk)$$

and $n$ is a smooth kernel satisfying an integrability condition. We assume that $\beta, \alpha \in C^{\eta}(\mathbb{R}^d)$ with $\eta \in (0,1]$, and $\alpha > 0$. Under appropriate spectral theoretical assumptions we prove the existence of the random measure $$\lim_{t \uparrow \infty} e^{-\lambda_c t} X_t (dx)$$ (with respect to the vague topology), where $\lambda_c$ is the generalized principal eigenvalue of $L + \beta$ on $\mathbb{R}^d$ 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

$$\lim_{t \uparrow \infty} e^{-\beta t} X_t (dx)$$

exists and equals a nonnegative nondegenerate random multiple of the invariant measure for $Y$.

Taking $L = 1/2 \Delta$ on $\mathbb{R}$ and replacing $\beta$ by $\delta_0$ (super-Brownian motion with a single point source), we prove a similar result with $\lambda_c$ replaced by 1/2 and with the deterministic measure $e^{-|x| dx$, giving an answer in theaffirmative 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.