Abstract
In this article we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Our methods rely on local central limit theorems for convergence of random walks on discretizations of smooth domains to Reflected Brownian motion.
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References
Ahlfors, L.V.: Complex analysis: an introduction to the theory of analytic functions of one complex variable (1979)
Bass, R.F.: Brownian motion, heat kernels, and harmonic functions. In: Proceedings of the International Congress of Mathematicians, pp 980–985. Springer (1995)
Bass, R.F., Hsu, P.: Some potential theory for reflecting Brownian motion in holder and lipschitz domains. Ann. Probab., 486–508 (1991)
Begehr, H.: Boundary value problems in complex analysis ii. Boletın de la asociación matemática Venezolana, p 217 (2005)
Burdzy, K., Chen, Z.-Q.: Discrete approximations to reflected Brownian motion. Ann. Probab. 36(2), 698–727 (2008)
Chen, Z.-Q.: On reflecting diffusion processes and skorokhod decompositions. Probab. Theory Relat. Fields 94(3), 281–315 (1993)
Chen, Z.-Q., Fan,W.-T.: Hydrodynamic limits and propagation of chaos for interacting random walks in domains. Ann. Appl. Probab. to appear (2016). arXiv:1311.2325
Fan, W.-T.: Interacting particle systems with partial annihilation through membranes. Phd thesis University of Washington (2014)
Ganguly, S., Peres, Y.: Competitive erosion is conformally invariant. arXiv:1503.06989 (2015)
Kesten, H.: Relations between solutions to a discrete and continuous Dirichlet problem. In: Random Walks, Brownian Motion, and Interacting Particle Systems, pp 309–321. Springer (1991)
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI (2009)
Morris, B., Peres, Y.: Evolving sets, mixing and heat kernel bounds. Probab. Theory Relat. Fields 133(2), 245–266 (2005)
Pommerenke, C.: Boundary behaviour of conformal maps. Springer (1992)
Sato, K.-I., Ueno, T.: Multi-dimensional diffusion and the markov process on the boundary. Journal of Mathematics of Kyoto University 4(3), 529–605 (1965)
Smirnov, S.: Discrete complex analysis and probability. arXiv:1009.6077 (2010)
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Ganguly, S., Peres, Y. Convergence of Discrete Green Functions with Neumann Boundary Conditions. Potential Anal 46, 799–818 (2017). https://doi.org/10.1007/s11118-016-9602-x
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DOI: https://doi.org/10.1007/s11118-016-9602-x