Law of iterated logarithm for supercritical symmetric branching Markov process

Abstract: Let ${(X_t)_{t\geq 0}, \mathbb{P}_x, x\in E}$ be a supercritical symmetric branching Markov process 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, 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 $f\in T_r(L^2(E, \mu))$ for some $r>0$.