Law of iterated logarithm for supercritical non-symmetric branching Markov process

Let $\{(X_t)_{t\geq 0}, \mathbb{P}_{δ_x}, x\in E\}$ be a supercritical branching Markov process (which is not necessary symmetric) on a locally compact metric measure space $(E,μ)$ 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. ; 49 pages