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

J. Shin

School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea

Abstract: 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 $(X_t)_{t \ge 0}$.

Keywords: semimartingale decomposition, Dirichlet forms, reflected stable-like processes, heat kernel estimates, Hölder continuity.

Received: 31.10.2017
Accepted: 12.02.2019

DOI: 10.4213/tvp5171

 English version: Theory of Probability and its Applications, 2019, 64:3, 421–443

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