Subcritical superprocesses conditioned on non-extinction

Abstract

We consider a class of subcritical superprocesses $(X_t{)}_{t\ge 0}$ with general spatial motions and general branching mechanisms. We study the asymptotic behaviors of ${\mathbf{Q}}_{t,r}$, the distribution of $X_t$ conditioned on $X_{t+r}$ not being a null measure. We first give the existence of ${\lim }_{t\to \infty }{\mathbf{Q}}_{t,r}$ and ${\lim }_{r\to \infty }{\mathbf{Q}}_{t,r}$, and then show that an $L\log L$-type condition is equivalent to the existence of the double limits: ${\lim }_{r\to \infty }{\lim }_{t\to \infty }{\mathbf{Q}}_{t,r}$ and ${\lim }_{t\to \infty }{\lim }_{r\to \infty }{\mathbf{Q}}_{t,r}$. Finally, when the $L\log L$-type condition holds, we show that those double limits, and ${\lim }_{r,t\to \infty }{\mathbf{Q}}_{t,r}$, are the same.

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