Subcritical superprocesses conditioned on non-extinction

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Abstract: We consider a class of subcritical superprocesses

(X_{t})_{t g e q 0}

with general spatial motions and general branching mechanisms. We study the asymptotic behaviors of

m a t h b f Q_{t, r}

, the distribution of

X_{t}

conditioned on

X_{t + r}

not being a null measure. We first give the existence of

l i m_{t o i n f t y} m a t h b f Q_{t, r}

and

l i m_{r o i n f t y} m a t h b f Q_{t, r}

, and then show that an

L l o g L

-type condition is equivalent to the existence of the double limits:

l i m_{r o i n f t y} l i m_{t o i n f t y} m a t h b f Q_{t, r}

and

l i m_{t o i n f t y} l i m_{r o i n f t y} m a t h b f Q_{t, r}

. Finally, when the

L l o g L

-type condition holds, we show that those double limits, and

l i m_{r, t o i n f t y} m a t h b f Q_{t, r}

, are the same.

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