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A graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold

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Abstract

In the present paper, we prove that the \(C_{0}\)-semigroup generated by a Schrödinger operator with drift on a complete Riemannian manifold is approximated by the discrete semigroups associated with a family of discrete time random walks with killing in a flow on a sequence of proximity graphs, which are constructed by partitions of the manifold. Furthermore, when the manifold is compact, we also obtain a quantitative error estimate of the convergence. Finally, we give examples of the partition of the manifold and the drift term on two typical manifolds: Euclidean spaces and model manifolds.

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Acknowledgements

The authors are grateful to Professor Atsushi Kasue for useful discussions on the Laplacian comparison theorem. They also thank Professors Atsushi Atsuji, Batu Güneysu, Jun Masamune, Ryuya Namba and Ichiro Shigekawa for giving valuable comments. The first author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 17K05215, (C) No. 22K03280 and (S) No. 22H04942. The second author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 17K05300, (C) No. 20K03639 and (B) No. 21H00988.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Science, Yamagata University, 1-4-12, Kojirakawa, Yamagata, 990-8560, Japan

    Satoshi Ishiwata

  2. Department of Mathematics, Hiyoshi Campus, Keio University, 4-1-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8521, Japan

    Hiroshi Kawabi

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  1. Satoshi Ishiwata
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Correspondence to Satoshi Ishiwata.

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Ishiwata, S., Kawabi, H. A graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02809-9

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