Weak extinction versus global exponential growth of total mass for superdiffusions

الأصل · EN

Consider a superdiffusion X on Rᵈ corresponding to the semilinear operator A(u)=Lu+βu-ku², where L is a second order elliptic operator, β(·) is in the Kato class and bounded from above, and k(·)≥ 0 is bounded on compact subsets of ᵈ and is positive on a set of positive Lebesgue measure. The main purpose of this paper is to complement the results obtained in Englander:2004, in the following sense. Let λ∞ be the L∞-growth bound of the semigroup corresponding to the Schrödinger operator L+β. If λ∞ ≠0, then we prove that, in some sense, the exponential growth/decay rate of Xₜ, the total mass of Xₜ, is λ∞. We also describe the limiting behavior of (-λ∞ t)Xₜ in these cases. This should be compared to the result in Englander:2004, 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 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 local extinction (which only depends on the sign of λ₂, and which turns out to be equivalent to local extinction) of X.