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.