  In this paper we establish spatial central limit theorems for a large class
of supercritical branching Markov processes with general spatial-dependent
branching mechanisms. These are generalizations of the spatial central limit
theorems proved in [1], P. for branching OU processes with binary branching
mechanisms. Compared with the results of [1], our central limit theorems are
more satisfactory in the sense that the normal random variables in our theorems
are non-degenerate.
