Weak extinction versus global exponential growth of total mass for superdiffusions

Consider a superdiffusion X on ℝ^d corresponding to the semi-linear operator A(u) = Lu + βu-ku², where L is a second order elliptic operator, β(.) is in the Kato class, and k(.) ≥ 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 (Ann. Probab. 32 (2004) 78.99), in the following sense. Let λ be the L-growth bound of the semigroup corresponding to the Schrodinger-type operator L + β. If λ ≠ 0, then we prove that, in some sense, the exponential growth/decay rate of ||X_t||, the total mass of X_t, is λ. We also describe the limiting behavior of exp(-λt)||X_t||, as t →. This should be compared to the result in (Ann. Probab. 32 (2004) 78-99), which says that the generalized principal eigenvalue λ₂ of the operator gives the rate of local growth when it is positive, and implies local extinction otherwise. It is easy to show that λ ≥ λ₂, and we discuss cases when λ > λ₂ and when λ = λ₂. When λ = 0, and under some conditions on β, we give a necessary and sufficient 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 local extinction. (The latter depends on the sign of λ₂ only, and it turns out to be equivalent to local extinction.)