zbMATH — the first resource for mathematics

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.
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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.