Limit theorems for branching Markov processes

We establish almost sure limit theorems for a branching symmetric Hunt process in terms of the principal eigenvalue and the ground state of an associated Schrödinger operator. Here the branching rate and the branching mechanism can be state-dependent. In particular, the branching rate can be a measure belonging to a certain Kato class and is allowed to be singular with respect to the symmetrizing measure for the underlying Hunt process X. The almost sure limit theorems are established under the assumption that the associated Schrödinger operator of X has a spectral gap. Such an assumption is satisfied if the underlying process X is a Brownian motion, a symmetric α-stable-like process on Rn or a relativistic symmetric stable process on Rn.