SEMIMARTINGALE DECOMPOSITION AND HEAT KERNEL ESTIMATES OF REFLECTED STABLE-LIKE PROCESSES WITH VARIABLE ORDER

摘要

We investigate symmetric reflected stable-like processes on a compact set (E) over bar subset of R-d associated to nonlocal Dirichlet forms with variable order alpha(. , .) in the jump intensity kernels. First, assuming two-sided estimates of the continuous transition density of the reflected stable-like process (X-t)(t >= 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 >= 0). Then by adding more conditions on alpha(. , .), we explicitly derive upper and lower bound estimates of the Holder continuous transition density of (X-t)(t >= 0).