Abstract
In this paper, we will show that if the sectional curvature of a Hadamard manifold $M$ is pinched by two negative constants, then $M$-valued jump-diffusion process $\{X_t;0\leq t\lt e\}$ satisfying suitable conditions on the Lévy measure is irreducible, transient and conservative. In order to show such properties of paths, we need the upper and lower estimates of the radial part of the jump-diffusion process.
Funding Statement
This work is supported by JST SPRING, Grant Number JPMJSP2139.
Acknowledgments
I would like to thank Professor Atsushi Takeuchi of Tokyo Woman's Christian University for his helpful discussion and encouragement. Professor Masamichi Yoshida of Osaka Metropolitan University gave me his support and important remarks on my research. I would also like to express my sincere gratitude to him. Professor Kazuhiro Kuwae of Fukuoka University gave very useful advice on this paper and introduced the relevant topics in terms of Dirichlet forms. I would like to express my gratitude to him as well.
Citation
Hirotaka Kai. "Long time behavior of jump-diffusion processes on Riemannian manifolds." Osaka J. Math. 61 (1) 25 - 52, January 2024.
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