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Two-dimensional Brownian random interlacements. (English) Zbl 1453.60136
Summary: We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [the authors and M. Vachkovskaia, Commun. Math. Phys. 343, No. 1, 129–164 (2016; Zbl 1336.60185)]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of the authors [Ann. Probab. 45, No. 6B, 4752–4785 (2017; Zbl 1409.60140)], the authors et al. [loc. cit.], as well as the results specific to the continuous case.
MSC:
60J45 Probabilistic potential theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J65 Brownian motion
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[1] Abe, Y.: Second order term of cover time for planar simple random walk arXiv:1709.08151 (2017)
[2] Baccelli, F.; Kim, KB; McDonald, D., Equilibria of a class of transport equations arising in congestion control, Queueing Syst., 55, 1, 1-8 (2007) · Zbl 1166.90323
[3] Baccelli, F., Carofiglio, G., Piancino, M.: Stochastic analysis of scalable TCP. In: Proceedings IEEE INFOCOM 2009, pp 19-27 (2009)
[4] Belius, D.; Kistler, N., The subleading order of two dimensional cover times, Probab. Theory Relat. Fields, 167, 1, 461-552 (2017) · Zbl 1365.60071
[5] Belius, D., Rosen, J., Zeitouni, O.: Tightness for the cover time of compact two dimensional manifolds. arXiv:1711.02845 (2017) · Zbl 1434.60222
[6] Camargo, D.; Popov, S., One-dimensional random interlacements, Stochastic Process. Appl., 128, 2750-2778 (2018) · Zbl 1388.60088
[7] Chen, H., Excursions in Classical Analysis (2010), Washington, DC: Mathematical Association of America, Washington, DC
[8] Chetrite, R.; Touchette, H., Nonequilibrium Markov processes conditioned on large deviations, Ann. Inst. Henri Poincaré B Probab. Stat., 16, 2005-2057 (2015) · Zbl 1330.82039
[9] Černý, J.; Teixeira, A., From random walk trajectories to random interlacements. Ensaios Matemáticos [Mathematical Surveys], vol. 23 (2012), Rio de Janeiro: Sociedade Brasileira de Matemática, Rio de Janeiro · Zbl 1269.60002
[10] Comets, F.; Popov, S., The vacant set of two-dimensional critical random interlacement is infinite, Ann. Probab., 45, 4752-4785 (2017) · Zbl 1409.60140
[11] Comets, F.; Gallesco, C.; Popov, S.; Vachkovskaia, M., On large deviations for the cover time of two-dimensional torus, Electr. J. Probab., 18, article 96 (2013) · Zbl 1294.60066
[12] Comets, F.; Popov, S.; Vachkovskaia, M., Two-dimensional random interlacements and late points for random walks, Commun. Math. Phys., 343, 129-164 (2016) · Zbl 1336.60185
[13] de Bernardini, DF; Gallesco, CF; Popov, S., On uniform closeness of local times of Markov chains and i.i.d. sequences, Stochastic Process. Appl., 128, 3221-3252 (2018) · Zbl 1409.60120
[14] Dembo, A.; Peres, Y.; Rosen, J.; Zeitouni, O., Cover times for Brownian motion and random walks in two dimensions, Ann. Math. (2), 160, 2, 433-464 (2004) · Zbl 1068.60018
[15] Dembo, A.; Peres, Y.; Rosen, J.; Zeitouni, O., Late points for random walks in two dimensions, Ann. Probab., 34, 1, 219-263 (2006) · Zbl 1100.60057
[16] Dereich, S., Döring, L.: Random interlacements via Kuznetsov measures. arXiv:1501.00649 (2015)
[17] Doob, JL, Classical Potential Theory and Its Probabilistic Counterpart (1984), Berlin: Springer, Berlin · Zbl 0549.31001
[18] Doob, JL, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85, 431-458 (1957) · Zbl 0097.34004
[19] Drewitz, A.; Ráth, B.; Sapozhnikov, A., An Introduction to Random Interlacements (2014), Berlin: Springer, Berlin · Zbl 1304.60008
[20] Goodman, J.; den Hollander, F., Extremal geometry of a Brownian porous medium, Probab. Theory Relat. Fields, 160, 1-2, 127-174 (2014) · Zbl 1327.60037
[21] Jacod, J.; Shiryaev, AN, Limit Theorems for Stochatic Processes. Grundlehren der mathematischen Wissenschaften (2003), Berlin: Springer, Berlin
[22] Kingman, JFC, Poisson Processes (1993), New York: Oxford University Press, New York
[23] Lawler, G.; Limic, V., Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, 123 (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1210.60002
[24] Lawler, GF; Werner, W., The Brownian loop soup, Probab. Theory Relat. Fields, 128, 4, 565-588 (2004) · Zbl 1049.60072
[25] Lawler, GF; Schramm, O.; Werner, W., Conformal restriction: the chordal case, J. Am. Math. Soc., 16, 4, 917-955 (2003) · Zbl 1030.60096
[26] Le Gall, JF, Some properties of planar Brownian motion. Ecole d’Été de Probabilités de Saint-Flour X-1990. Lecture Notes in Math., 1527, 111-235 (1992), Berlin: Springer, Berlin
[27] Le Jan, Y., Markov Paths, Loops and Fields (Saint-Flour Probability Summer School 2008). Lect. Notes Math. 2026 (2011), Berlin: Springer, Berlin
[28] Li, X.: Percolative properties of Brownian interlacements and its vacant set. arXiv:1610.08204 (2016)
[29] Li, X.; Sznitman, A-S, Large deviations for occupation time profiles of random interlacements, Probab. Theory Relat. Fields, 161, 1-2, 309-350 (2015) · Zbl 1314.60078
[30] Mörters, P.; Peres, Y., Brownian Motion (2010), Cambridge: Cambridge University Press, Cambridge
[31] Pitman, J., Yor, M.: Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In: Ikeda, N., Watanabe, S., Fukushima, M., Kunita, H. (eds.) Itô’s Stochastic Calculus and Probability Theory. Springer, Berlin (1996) · Zbl 0877.60053
[32] Popov, S.; Teixeira, A., Soft local times and decoupling of random interlacements, J. Eur. Math. Soc., 17, 10, 2545-2593 (2015) · Zbl 1329.60342
[33] Ransford, T., Potential Theory in the Complex Plane (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0828.31001
[34] Rodriguez, P.-F.: On pinned fields, interlacements, and random walk on \((\mathbb{Z}/N\mathbb{Z})^2 \). arXiv:1705.01934 To appear in: Probab. Theory Relat. Fields (2017)
[35] Rosen, J., Intersection local times for interlacements, Stochastic Process. Appl., 124, 5, 1849-1880 (2014) · Zbl 1307.60104
[36] Roynette, B., Yor, M.: Penalising Brownian Paths. Lect. Notes Math. 1969. Springer Science & Business Media (2009) · Zbl 1190.60002
[37] Sznitman, A-S, Vacant set of random interlacements and percolation, Ann. Math. (2), 171, 3, 2039-2087 (2010) · Zbl 1202.60160
[38] Sznitman, A-S, Topics in occupation times and Gaussian free fields. Zurich Lect. Adv. Math. (2012), Zürich: European Mathematical Society, Zürich · Zbl 1246.60003
[39] Sznitman, AS, On scaling limits and Brownian interlacements, Bull. Braz. Math. Soc. (N.S.), 44, 4, 555-592 (2013) · Zbl 1303.60022
[40] Teixeira, A., Interlacement percolation on transient weighted graphs, Electr. J. Probab., 14, 1604-1627 (2009) · Zbl 1192.60108
[41] Werner, W., Sur la forme des composantes connexes du complémentaire de la courbe brownienne plane, Probab. Theory Relat. Fields, 98, 3, 307-337 (1994) · Zbl 0792.60075
[42] Williams, D., Path decomposition and continuity of local time for one-dimensional diffusions. I, Proc. London Math. Soc., 28, 3, 738-768 (1974) · Zbl 0326.60093
[43] Wu, H.: On the occupation times of Brownian excursions and Brownian loops. Séminaire de Probabilités XLIV (Lect. Notes Math. 2046), pp. 149-166 (2012) · Zbl 1251.60062
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