Abstract
This chapter anticipates on later results that show how the limit theorems of Chaps. 5 and 6 apply to certain long-range random walks. It focuses on the problem of identifying the limit Lévy process that is obtained through these limit theorems when applied to a given explicit long-range random walk. This is done by drawing interesting links between variations on established results regarding approximations of Lévy processes on Lie groups on the one hand and the limit theorems obtained in Chaps. 5 and 6 on the other hand. Several examples are given to illustrate the limit theorems explicitly using this approach. Examples discussed in Chap. 1 are also revisited in this new light.
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References
Chen, Z.-Q., and M. Fukushima. 2012. Symmetric Markov processes, time change, and boundary theory. In London Mathematical Society Monographs Series, vol. 35. Princeton, NJ: Princeton University Press. MR 2849840
Fukushima, M., Y. Oshima, and M. Takeda. 2011. Dirichlet Forms and Symmetric Markov Processes, extended ed., De Gruyter Studies in Mathematics, vol. 19. Berlin: Walter de Gruyter & Co. MR 2778606
Kunita, H. 1994. Stable Lévy processes on nilpotent Lie groups. In Stochastic analysis on infinite-dimensional spaces (Baton Rouge, LA, 1994). Pitman Research Notes in Mathematics Series, vol. 310, 167–182. Harlow: Longman Scientific and Technical. MR 1415667
Kunita, H. 1995. Some problems concerning Lévy processes on Lie groups. In Stochastic Analysis (Ithaca, NY, 1993), Proceedings of Symposia in Pure Mathematics, vol. 57, 323–341. Providence, RI: American Mathematical Society. MR 1335479
Kurtz, T.G., and P.E. Protter. 1996. Weak convergence of stochastic integrals and differential equations. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Mathematics, vol. 1627, 1–41. Berlin: Springer. MR 1431298
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Chen, ZQ., Kumagai, T., Saloff-Coste, L., Wang, J., Zheng, T. (2023). Symmetric Lévy Processes on Nilpotent Groups. In: Limit Theorems for Some Long Range Random Walks on Torsion Free Nilpotent Groups. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-031-43332-0_7
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