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 Teor. Veroyatnost. i Primenen., 2019, Volume 64, Issue 3, Pages 526–551 (Mi tvp5171)

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.

DOI: https://doi.org/10.4213/tvp5171

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

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Accepted:12.02.2019

Citation: J. Shin, “Semimartingale decomposition and heat kernel estimates of reflected stable-like processes with variable order”, Teor. Veroyatnost. i Primenen., 64:3 (2019), 526–551; Theory Probab. Appl., 64:3 (2019), 421–443

Citation in format AMSBIB
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