Walking within growing domains: recurrence versus transience

Amir Dembo (Stanford University)
Ruojun Huang (Stanford University)
Vladas Sidoravicius (IMPA)

Abstract


For normally reflected Brownian motion and for simple random walk on independently growing in time $d$-dimensional domains, $d\ge3$, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-20

Publication Date: November 6, 2014

DOI: 10.1214/EJP.v19-3272

References

  • Amir, Gideon; Benjamini, Itai; Gurel-Gurevich, Ori; Kozma, Gady. Random walk in changing environment. Unpublished manuscript (2008).
  • Angel, Omer; Crawford, Nicholas; Kozma, Gady. Localization for linearly edge reinforced random walks. Duke Math. J. 163 (2014), no. 5, 889--921. MR3189433
  • Benjamini, Itai; Wilson, David B. Excited random walk. Electron. Comm. Probab. 8 (2003), 86--92 (electronic). MR1987097
  • Burdzy, Krzysztof; Chen, Zhen-Qing. Reflecting random walk in fractal domains. Ann. Probab. 41 (2013), no. 4, 2791--2819. MR3112932
  • Burdzy, Krzysztof; Chen, Zhen-Qing; Sylvester, John. The heat equation and reflected Brownian motion in time-dependent domains. Ann. Probab. 32 (2004), no. 1B, 775--804. MR2039943
  • Chen, Zhen-Qing; Croydon, David; Kumagai, Takashi. Quenched invariance principles for random walks and elliptic diffusions in random media with boundary. arXiv: 1306.0076v1 (2013). To appear in Ann. Probab.
  • Disertori, Margherita; Sabot, Christophe; Tarre, Pierre. Transience of edge-reinforced random walk. arXiv:1403.6079v2 (2014).
  • Dolgopyat, Dmitry; Keller, Gerhard; Liverani, Carlangelo. Random walk in Markovian environment. Ann. Probab. 36 (2008), no. 5, 1676--1710. MR2440920
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8 MR2722836
  • Fukushima, Masatoshi; Oshima, Yoichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. Second revised and extended edition. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 2011. x+489 pp. ISBN: 978-3-11-021808-4 MR2778606
  • Grimmett, G. R.; Kesten, H.; Zhang, Y. Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 (1993), no. 1, 33--44. MR1222363
  • Gyrya, Pavel; Saloff-Coste, Laurent. Neumann and Dirichlet heat kernels in inner uniform domains. Asterisque No. 336 (2011), viii+144 pp. ISBN: 978-2-85629-306-5 MR2807275
  • den Hollander, Frank; Molchanov, Stanislav A.; Zeitouni, Ofer. Random media at Saint-Flour. Reprints of lectures from the Annual Saint-Flour Probability Summer School held in Saint-Flour. Probability at Saint-Flour. Springer, Heidelberg, 2012. vi+564 pp. ISBN: 978-3-642-32948-7 MR3059554
  • Hsu, Pei. Brownian exit distribution of a ball. Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985), 108--116, Progr. Probab. Statist., 12, Birkhauser Boston, Boston, MA, 1986. MR0896739
  • Kesten, Harry. First-passage percolation. From classical to modern probability, 93--143, Progr. Probab., 54, Birkhauser, Basel, 2003. MR2045986
  • Kosygina, Elena; Zerner, Martin P. W. Excited random walks: results, methods, open problems. Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013), no. 1, 105--157. MR3097419
  • Kozma, Gady. Reinforced random walk. arXiv:1208.0364. Proc. of Europ. Cong. Math. (2012), 429-443.
  • Kozma, Gady. Centrally excited random walk is reccurent. Unpublished manuscript (2006).
  • Lawler, Gregory F. Intersections of random walks. Probability and its Applications. Birkhauser Boston, Inc., Boston, MA, 1991. 219 pp. ISBN: 0-8176-3557-2 MR1117680
  • Lawler, Gregory F.; Bramson, Maury; Griffeath, David. Internal diffusion limited aggregation. Ann. Probab. 20 (1992), no. 4, 2117--2140. MR1188055
  • Morters, Peter; Peres, Yuval. Brownian motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. xii+403 pp. ISBN: 978-0-521-76018-8 MR2604525
  • Pascu, Mihai N. Mirror coupling of reflecting Brownian motion and an application to Chavel's conjecture. Electron. J. Probab. 16 (2011), No. 18, 504--530. MR2781844
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Sabot, Christophe; Tarres, Pierre. Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. arXiv:1111.3991v4 (2012). To appear J. Eur. Math. Soc.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.