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The High-Order Corrections of Discrete Harmonic Measures and Their Correction Constants

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Abstract

By the dimension reduction idea, overshoot for random walks, coupling and martingale arguments, we obtain a simpler and easily computable expression for the first-order correction constant between discrete harmonic measures for random walks with rotationally invariant step distribution in \(\mathbb {R}^d\ (d\ge 2)\) and the corresponding continuous counterparts. This confirms and extends a conjecture in Jiang and Kennedy (J Theor Probab 30(4):1424–1444, 2017), and simplifies the related expression of Wang et al. (Bernoulli 25(3):2279–2300, 2019). Furthermore, we propose a universality conjecture on high-order corrections for error estimation between generalized discrete harmonic measures and their continuous counterparts, which generalizes the universality conjecture of the first-order correction in Kennedy (J Stat Phys 164(1):174–189, 2016); and we prove this conjecture heuristically for the rotationally invariant case, and also provide several examples of second-order error corrections to check the conjecture by a numerical simulation argument.

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Authors and Affiliations

  1. National Center for Applied Mathematics in Hunan, Xiangtan University, Xiangtan, 411105, Hunan Province, People’s Republic of China

    Yixiang Wang, Kainan Xiang & Lang Zou

  2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan Province, People’s Republic of China

    Yixiang Wang, Kainan Xiang & Lang Zou

  3. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, CEP 05508-090, Brazil

    Shangjie Yang

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  1. Yixiang Wang
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  2. Kainan Xiang
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  4. Lang Zou
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Correspondence to Lang Zou.

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Communicated by Hal Tasaki.

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The project is supported partially by the NSFC (No. 12171410, No. 12001463), the Education Department of Hunan Province (No. 23B0178), Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai (No. 2019RS1057) and the Graduate Innovation Project of Xiangtan University (No. XDCX2023Y105). S. Yang is supported by FAPSP 2023/12652-4

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Wang, Y., Xiang, K., Yang, S. et al. The High-Order Corrections of Discrete Harmonic Measures and Their Correction Constants. J Stat Phys 191, 81 (2024). https://doi.org/10.1007/s10955-024-03292-x

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