Law of large numbers for branching symmetric Hunt processes with measure-valued branching rates

We establish weak and strong law of large numbers for a class of branching symmetric Hunt processes with the branching rate being a smooth measure with respect to the underlying Hunt process, and the branching mechanism being general and state-dependent. Our work is motivated by recent work on strong law of large numbers for branching symmetric Markov processes by Chen-Shiozawa [J. Funct. Anal., 250, 374–399, 2007] and for branching diffusions by Engl"ander-Harris-Kyprianou [Ann. Inst. Henri Poincar'e Probab. Stat., 46, 279–298, 2010]. Our results can be applied to some interesting examples that are covered by neither of these papers.