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

Let $\{(X_t{)}_{t\ge 0},{\mathbb{P}}_{{\delta }_x},x\in E\}$ 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 $⟨f,X_t⟩$ under the second moment condition on the branching mechanism, where $f$ is a linear combination of eigenfunctions of the mean semigroup $\{T_t,t\ge 0\}$ of $X$. Then we prove law of iterated logarithm type results for $⟨f,X_t⟩$ under the fourth moment condition, where $f$ belongs to a larger class of functions.