• Download PDF
• Download TeX
• Download bibTeX entry

Metadata

Abstract

References

[AD01] Alili, Larbi; Doney, Ronald A. Martin boundaries associated with a killed random walk, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 37 (2001) no. 3, pp. 313-338 | DOI | Numdam | MR | Zbl

[AGKV05] Afanasyev, Valeriĭ I.; Geiger, Jochen; Kersting, Götz-Dietrich; Vatutin, Vladimir A. Criticality for branching processes in random environment, Ann. Probab., Volume 33 (2005) no. 2, pp. 645-673 | MR | Zbl

[BBMKM16] Bostan, Alin; Bousquet-Mélou, Mireille; Kauers, M.; Melczer, Stephen On $3$-dimensional lattice walks confined to the positive octant, Ann. Comb., Volume 20 (2016) no. 4, pp. 661-704 | DOI | MR | Zbl

[BBMM21] Bostan, Alin; Bousquet-Mélou, Mireille; Melczer, Stephen Counting walks with large steps in an orthant, J. Eur. Math. Soc. (JEMS), Volume 23 (2021) no. 7, pp. 2221-2297 | DOI | MR | Zbl

[BBO05] Biane, Philippe; Bougerol, Philippe; O’Connell, Neil Littelmann paths and Brownian paths, Duke Math. J., Volume 130 (2005) no. 1, pp. 127-167 | MR | Zbl

[Bia91] Biane, Philippe Quantum random walk on the dual of SU$\left(n\right)$, Probab. Theory Relat. Fields, Volume 89 (1991) no. 1, pp. 117-129 | DOI | MR | Zbl

[Bia92] Biane, Philippe Équation de Choquet–Deny sur le dual d’un groupe compacts, Probab. Theory Relat. Fields, Volume 94 (1992) no. 1, pp. 39-51 | DOI | Zbl

[BMM10] Bousquet-Mélou, Mireille; Mishna, Marni, Algorithmic probability and combinatorics. Papers from the AMS special sessions, Chicago, IL, USA, October 5–6, 2007 and Vancouver, BC, Canada, October 4–5, 2008 (Contemporary Mathematics), Volume 520 (2010), pp. 1-39 | MR | Zbl

[BMS15] Bouaziz, Aymen; Mustapha, Sami; Sifi, Mohamed Discrete harmonic functions on an orthant in ${\mathbb{Z}}^d$, Electron. Commun. Probab., Volume 20 (2015), 52, 13 pages | MR | Zbl

[BS97] Bañuelos, Rodrigo; Smits, Robert G. Brownian motion in cones, Probab. Theory Relat. Fields, Volume 108 (1997) no. 3, pp. 299-319 | DOI | MR | Zbl

[BT12] Billiard, Sylvain; Tran, Viet Chi A general stochastic model for sporophytic self-incompatibility, J. Math. Biol., Volume 64 (2012) no. 1-2, pp. 163-210 | DOI | MR | Zbl

[CB83] Cohen, Jacob W.; Boxma, Onno J. Boundary value problems in queueing system analysis, North-Holland Mathematics Studies, 79, North-Holland, 1983 | Zbl

[CD19] Caravenna, Francesco; Doney, Ronald A. Local large deviations and the strong renewal theorem, Electron. J. Probab., Volume 24 (2019), 72, 48 pages | MR | Zbl

[CdL13] Cont, Rama; de Larrard, Adrien Price dynamics in a Markovian limit order market, SIAM J. Financial Math., Volume 4 (2013), pp. 1-25 | DOI | MR | Zbl

[DHRS18] Dreyfus, Thomas; Hardouin, Charlotte; Roques, Julien; Singer, Michael F. On the nature of the generating series of walks in the quarter plane, Invent. Math., Volume 213 (2018) no. 1, pp. 139-203 | DOI | MR | Zbl

[DIM77] Durrett, Richard T.; Iglehart, Donald L.; Miller, Douglas R. Weak convergence to Brownian meander and Brownian excursion, Ann. Probab., Volume 5 (1977), pp. 117-129 | MR | Zbl

[Don98] Doney, Ronald A. The Martin boundary and ratio limit theorems for killed random walks, J. Lond. Math. Soc., Volume 58 (1998) no. 3, pp. 761-768 | DOI | MR | Zbl

[Don12] Doney, Ronald A. Local behaviour of first passage probabilities, Probab. Theory Relat. Fields, Volume 152 (2012) no. 3-4, pp. 559-588 | DOI | MR | Zbl

[DSW18] Denisov, Denis; Sakhanenko, Alexander; Wachtel, Vitali First-passage times for random walks with nonidentically distributed increments, Ann. Probab., Volume 46 (2018) no. 6, pp. 3313-3350 | MR | Zbl

[Dur78] Durrett, Richard T. Conditioned limit theorems for some null recurrent Markov processes, Ann. Probab., Volume 6 (1978), pp. 798-828 | MR | Zbl

[DW10] Denisov, Denis; Wachtel, Vitali Conditional limit theorems for ordered random walks, Electron. J. Probab., Volume 15 (2010), 11, pp. 292-322 | MR | Zbl

[DW15] Denisov, Denis; Wachtel, Vitali Random walks in cones, Ann. Probab., Volume 43 (2015) no. 3, pp. 992-1044 | MR | Zbl

[DW16] Denisov, Denis; Wachtel, Vitali An exact asymptotics for the moment of crossing a curved boundary by an asymptotically stable random walk, Theory Probab. Appl., Volume 60 (2016) no. 3, pp. 481-500 | MR | Zbl

[DW19] Denisov, Denis; Wachtel, Vitali Alternative constructions of a harmonic function for a random walk in a cone, Electron. J. Probab., Volume 24 (2019), 92, 26 pages | MR | Zbl

[DW20] Duraj, Jetlir; Wachtel, Vitali Invariance principles for random walks in cones, Stochastic Processes Appl., Volume 130 (2020) no. 7, pp. 3920-3942 | DOI | MR | Zbl

[EK08] Eichelsbacher, Peter; König, W. Ordered random walks, Electron. J. Probab., Volume 13 (2008), pp. 1307-1336 | MR | Zbl

[Ess68] Esseen, Carl-Gustav On the concentration function of a sum of independent random variables, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 9 (1968), pp. 290-308 | DOI | MR | Zbl

[FIM17] Fayolle, Guy; Iasnogorodski, Roudolf; Malyshev, Vadim Random walks in the quarter plane. Algebraic methods, boundary value problems, applications to queueing systems and analytic combinatorics, Probability Theory and Stochastic Modelling, 40, Springer, 2017 (second edition, previously published with the subtitle Algebraic methods, boundary value problems and applications) | Zbl

[FN71] Fuk, Dao Ha; Nagaev, Sergey V. Probabilistic inequalities for sums of independent random variables, Theory Probab. Appl., Volume 16 (1971) no. 4, pp. 643-660 | MR | Zbl

[GSC11] Gyrya, Pavel; Saloff-Coste, Laurent Neumann and Dirichlet heat kernels in inner uniform domains, Astérisque, 336, Société Mathématique de France, 2011 | Numdam | Zbl

[GT01] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001 | DOI | Zbl

[GZ09] Götze, Friedrich; Zaĭtsev, Andrei Yu The accuracy of approximation in the multidimensional invariance principle for sums of independent identically distributed random vectors with finite moments, Zap. Nauchn. Semin. (POMI), Volume 368 (2009) no. 15, p. 110-121, 283–284 | DOI

[HUL01] Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude Fundamentals of Convex Analysis, Grundlehren. Text Editions, 305-306, Springer, 2001 | DOI | Zbl

[IR08] Ignatiouk-Robert, Irina Martin boundary of a killed random walk on a half-space, J. Theor. Probab., Volume 21 (2008) no. 1, pp. 35-68 | DOI | MR | Zbl

[IR09] Ignatiouk-Robert, Irina Martin boundary of a killed random walk on ${\mathbb{Z}}_{+}^d$ (2009) (https://arxiv.org/abs/0909.3921)

[IR21] Ignatiouk-Robert, Irina Harmonic functions of random walks in a semigroup via ladder heights, J. Theor. Probab., Volume 34 (2021) no. 1, pp. 34-80 | DOI | MR | Zbl

[IRL10] Ignatiouk-Robert, Irina; Loree, Christophe Martin boundary of a killed random walk on a quadrant, Ann. Probab., Volume 38 (2010) no. 3, pp. 1106-1142 | MR | Zbl

[LR16] Lecouvey, Cédric; Raschel, Kilian $t$-Martin boundary of killed random walks in the quadrant, Séminaire de Probabilités XLVIII (Lecture Notes in Mathematics), Volume 2168, Springer, 2016, pp. 305-323 | DOI | MR | Zbl

[Mog73] Mogul’skii, Anatolii A. Absolute estimates for moments of certain boundary functionals, Theory Probab. Appl., Volume 18 (1973), pp. 340-347 | DOI | MR | Zbl

[MS19] Mustapha, Sami; Sifi, Mohamed Discrete harmonic functions in Lipschitz domains, Electron. Commun. Probab., Volume 24 (2019), 58, 15 pages | MR | Zbl

[NS66] Ney, Peter E.; Spitzer, Frank L. The Martin boundary for random walk, Trans. Am. Math. Soc., Volume 121 (1966), pp. 116-132 | MR | Zbl

[PW92] Picardello, Massimo A.; Woess, Wolfgang Martin boundaries of Cartesian products of Markov chains, Nagoya Math. J., Volume 128 (1992), pp. 153-169 | DOI | MR | Zbl

[Ras11] Raschel, Kilian Green functions for killed random walks in the Weyl chamber of Sp$\left(4\right)$, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 47 (2011) no. 4, pp. 1001-1019 | Numdam | MR | Zbl

[Ras14] Raschel, Kilian Random walks in the quarter plane, discrete harmonic functions and conformal mappings. With an appendix by S. Franceschi, Stochastic Processes Appl., Volume 124 (2014) no. 10, pp. 3147-3178 | DOI | Zbl

[RT20] Raschel, Kilian; Tarrago, Pierre Boundary behavior of random walks in cones, Markov Process. Relat. Fields, Volume 26 (2020) no. 4, pp. 711-756 | MR | Zbl

[RY05] Revuz, Daniel; Yor, Marc Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften, 293, Springer, 2005 | Zbl

[Saw97] Sawyer, Stanley A. Martin boundaries and random walks, Harmonic functions on trees and buildings. Workshop on harmonic functions on graphs, New York, NY, October 30–November 3, 1995 (Contemporary Mathematics), Volume 206, American Mathematical Society, 1997, pp. 17-44 | MR | Zbl

[Spi76] Spitzer, Frank L. Principles of Random Walk, Graduate Texts in Mathematics, 34, Springer, 1976 | DOI | MR | Zbl

[Ste90] Stembridge, John R. Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math., Volume 83 (1990) no. 1, pp. 96-131 | DOI | MR | Zbl

[Uch98] Uchiyama, Kôhei Green’s functions for random walks on ${\mathbb{Z}}^N$, Proc. Lond. Math. Soc., Volume 77 (1998) no. 1, pp. 215-240 | DOI | MR | Zbl

[Uch14] Uchiyama, Kôhei Green’s functions of random walks on the upper half plane, Tôhoku Math. J., Volume 66 (2014) no. 2, pp. 289-307 | MR | Zbl

[Var99] Varopoulos, Nicolas Th. Potential theory in conical domain, Math. Proc. Camb. Philos. Soc., Volume 125 (1999) no. 2, pp. 335-384 | DOI | MR | Zbl

[Var09] Varopoulos, Nicolas Th. The discrete and classical Dirichlet problem, Milan J. Math., Volume 77 (2009), pp. 397-436 | DOI | MR | Zbl

[Wil68] Williamson, John A. Random walks and Riesz kernels, Pac. J. Math., Volume 25 (1968), pp. 393-415 | DOI | MR | Zbl