Weak extinction versus global exponential growth of total mass for superdiffusions

Consider a superdiffusion $X$ on $\mathbb R^d$ corresponding to the semilinear operator $\mathcal{A}(u)=Lu+βu-ku^2,$ where $L$ is a second order elliptic operator, $β(\cdot)$ is in the Kato class and bounded from above, and $k(\cdot)\ge 0$ is bounded on compact subsets of $\R^d$ and is positive on a set of positive Lebesgue measure. The main purpose of this paper is to complement the results obtained in \cite{Englander:2004}, in the following sense. Let $λ_\infty$ be the $L^\infty$-growth bound of the semigroup corresponding to the Schrödinger operator $L+β$. If $λ_\infty \neq0$, then we prove that, in some sense, the exponential growth/decay rate of $\|X_t\|$, the total mass of $X_t$, is $λ_\infty$. We also describe the limiting behavior of $\exp(-λ_\infty t)\|X_t\|$ in these cases. This should be compared to the result in \cite{Englander:2004}, which says that the generalized principal eigenvalue $λ_2$ of the operator gives the rate of {\it local} growth when it is positive, and implies local extinction otherwise. It is easy to show that $λ_{\infty}\ge λ_2$, and we discuss cases when $λ_{\infty}> λ_2$ and when $λ_{\infty}= λ_2$. When $λ_\infty =0$, and under some conditions on $β$, we give a sufficient and necessary condition for the superdiffusion $X$ to exhibit weak extinction. We show that the branching intensity $k$ affects weak extinction; this should be compared to the known result that $k$ does not affect weak {\it local} extinction (which only depends on the sign of $λ_2$, and which turns out to be equivalent to local extinction) of $X$.

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