Semimartingale decomposition and heat kernel estimates of reflected stable-like processes with variable order

We investigate symmetric reflected stable-like processes on a compact set $ \overline{E} \subset \mathbf{R}^d$ associated to nonlocal Dirichlet forms with variable order $\alpha{(\,\cdot\,,\cdot\,)}$ in the jump intensity kernels. First, assuming two-sided estimates of the continuous transition density of the reflected stable-like process $(X_t)_{t \ge 0}$, similarly to [Q.-Y. Guan and Z.-M. Ma, Probab. Theory Related Fields, 134 (2006), pp. 649–694], we obtain the semimartingale decomposition of the process $(X_t)_{t \ge 0}$. Then by adding more conditions on $\alpha{(\,\cdot\,,\cdot\,)}$, we explicitly derive upper and lower bound estimates of the Hölder continuous transition density of $(X_t)_{t \ge 0}$.