Abstract:

Suppose X is a superdiffusion on domain $D\subseteq\textrm{R}^{d}$, whose underlying motion ζ is a diffusion process with absorption at the boundary corresponding to generator
 $$L=\frac{1}{2}\sum\limits_{i,j=1}^{d}a_{i,j}(x)\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}+\sum\limits_{i=1}^{d}b_{i}(x)\frac{\partial}{\partial x_{i}},$$
 whose  branching rate function is dt, and branching mechanism is of the form ψ(x, z) =α(x)z², x ∈D, α∈C^η(D)(0<η≤1), and α(x)≡0 on D₀(D₀ is a bounded domain in D), α>0 on D\D₀. The authors prove criteria for extinction, compact support property and global support being compact.

Key words: Superdiffusion, Extinction, Compact support property, Global support being compact