Let $\{ (X_t)_{ t\geq 0}, \mathbb{ P}_{ \delta_x}, x\in E\}$ be a supercritical branching Markov process (which is not necessary symmetric) on a locally compact metric measure space $(E,\mu)$ with spatially dependent local branching mechanism. Under some assumptions on the semigroup of the spatial motion, we first prove law of iterated logarithm type results for $\langle f, X_t\rangle$ under the second moment condition on the branching mechanism, where $f$ is a linear combination of eigenfunctions of the mean semigroup $\{ T_t, t\geq0\}$ of $X$. Then we prove law of iterated logarithm type results for $\langle f, X_t\rangle$ under the fourth moment condition, where $f$ belongs to a larger class of functions.
Categories: math.PR