Abstract
Firstly, we shall introduce the so-called snapping out Walsh’s Brownian motion and present its relation with Walsh’s Brownian motion. Then the stiff problem related to Walsh’s Brownian motion will be described and we shall build a phase transition for it. The snapping out Walsh’s Brownian motion corresponds to the so-called semi-permeable pattern of this stiff problem.
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References
Barlow, M., Pitman, J., Yor, M.: On Walsh’s Brownian motions. In: Séminaire de Probabilités, XXIII, Lecture Notes in Math., vol. 1372, pp. 275–293. Springer, Berlin (1989), https://doi.org/10.1007/BFb0083979
Chen, Z.Q., Fukushima, M.: One-point extensions of Markov processes by darning. Probab. Theory Relat. Fields 141(1–2), 61–112 (2008)
Chen, Z.Q., Fukushima, M.: Symmetric Markov processes, time change, and boundary theory. In: London Mathematical Society Monographs Series, vol. 35. Princeton University Press, Princeton (2012)
Chen, Z.Q., Fukushima, M.: One-point reflection. Stochastic Process. Appl. 125(4), 1368–1393 (2015). https://doi.org/10.1016/j.spa.2014.11.002
Chen, Z.Q., Peng, J.: Markov processes with darning and their approximations. Stochastic Process. Appl. 128(9), 3030–3053 (2018). https://doi.org/10.1016/j.spa.2017.10.009
Chen, Z.Q., Fukushima, M., Ying, J.: Traces of symmetric Markov processes and their characterizations. Ann. Probab. 34 (3), 1052–1102 (2006). https://doi.org/10.1214/009117905000000657
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, vol. 19, extended edn. Walter de Gruyter & Co., Berlin (2011)
Itô, K., McKean, H.P. Jr.: Diffusion Processes and Their Sample Paths. Springer, Berlin-New York (1974). Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125
Lejay, A.: The snapping out Brownian motion. Ann. Appl. Probab. 26(3), 1727–1742 (2016). https://doi.org/10.1214/15-AAP1131
Li, L., Sun, W.: On stiff problems via dirichlet forms. arXiv:1804.02634v3
Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2), 368–421 (1994)
Sanchez-Palencia, E.: Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin (1980)
Walsh, J.B.: A diffusion with a discontinuous local time. Astérisque 52–53, 37–45 (1978)
Acknowledgements
The first named author is partially supported by NSFC (No. 11688101 and 11801546) and Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (No. 2008DP173182). The second named author is partially supported by China Scholarship Council (No. 201706100095).
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Li, L., Sun, W. Snapping Out Walsh’s Brownian Motion and Related Stiff Problem. Potential Anal 53, 113–130 (2020). https://doi.org/10.1007/s11118-018-09761-9
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DOI: https://doi.org/10.1007/s11118-018-09761-9