- W. Woess:
*Denumerable Markov Chains - Generating Functions, Boundary Theory, Random Walks on Trees*, EMS Textbooks in Mathematics, European Mathematical Society Publishing House, xviii+351 pages, 2009.

Book reviews: Mathematical Reviews (pdf), Zentralblatt (pdf). Individual feedback (pdf)

Once a book is published, as the author one is bound to find some mistake or misprint every time one opens it. After resisting to open it for quite a while, in April 2010 I found one such mistake: on page 190, the random time "bold epsilon" is not a stopping time. However, this wrong statement is irrelevant for what follows. Another little misprint was pointed out to my by Irving Glick (May, 2010): on page 16, last line, the sum should start with n=1. Eleven years later: page 143, line 6 from bottom: "= y" instead of "= x". Page 144, line 4: it should be "produces k children". Page 149, in the proof of Theorem 5.54: it should be u(m) = 11...12, not 00...01.

In April 2023, Jun Kigami pointed out a mistake in the proof of Proposition 1.43. A corrected version can be found here (pdf).

Regarding the potential theoretic issues of Section 6, I found it desirable to provide some additional remarks on the*Riesz decomposition*for superharmonic functions that are not necessarily bounded below (pdf file). - W. Woess:
*Random Walks on Infinite Graphs and Groups*, Cambridge Tracts in Mathematics 138, Cambridge University Press, 334+xi pages, 2000. Paperback re-edition, 2008.

ERRATA - Four pages of "errata" regarding various misprints and three "true" mistakes. Last updated in January, 2015.

Book reviews: Mathematical Reviews (pdf), Zentralblatt (pdf), Bull. London Math. Soc. (pdf; attention - text starts on bottom of page); Acta Sci. Math. (Szeged) (pdf), Bulletin Amer. Math. Soc. (pdf), Nieuw Archief voor Wiskunde (pdf).

You can find a CORRECTED VERSION (2022) here! - W. Woess:
*Catene di Markov e Teoria del Potenziale nel Discreto*, Quaderni dell' Unione Matematica Italiana**41**, 165+iii pages, 1996.

Book review: Zentralblatt (pdf). - D. Lenz, F. Sobieczky and W. Woess (editors):
*Random Walks, Boundaries and Spectra*, Proceedings (Graz - St. Kathrein, 2009), Progess in Probability, vol 64. Birkhäuser, Basel, 324+xxvi pages, 2011.

- V. A. Kaimanovich with K. Schmidt and W. Woess (editors):
*Random Walks and Geometry*, Proceedings (ESI, Vienna, 2001), de Gruyter, Berlin, 532+x pages, 2004. - P. M. A. Grabner, W. Woess (editors):
*Fractals in Graz 2001: Analysis - Dynamics - Geometry - Stochastics*, Proceedings (Graz 2001), Birkhäuser, Basel, 283+iii pages, 2003. - M. A. Picardello, W. Woess (editors):
*Random Walks and Discrete Potential Theory*, Proceedings (Cortona 1997), Symposia Mathematica**XXXIX**, Cambridge University Press, 361+ix pages, 1999.

All downloads are pdf. The "Remark"s appearing in the list are popups containing some personal background.

- W. Woess:
*Notes on hyperbolic branching Brownian motion*, work in progress, posted here on April 9, 2024. Remark - M. Salvatori, M. A. Picardello, W. Woess:
*Polyharmonic potential theory on the Poincaré disk*, Journal of Functional Analysis 286, Issue 9 (2024) article no. 110362. - R. I. Möller, W. Woess:
*Amalgamated products of G-spaces*, unfinished manuscript, TU Graz (2020/22). Remark - W. Woess:
*Some old and basic facts about random walks on groups*, arXiv:2209.00319 (2022). Remark - V. A. Kaimanovich and W. Woess:
*Limit distributions of branching Markov chains*, Annales Inst. H. Poincaré - Prob. & Stat. 59, No 4 (2023) 1951-1983. - A. Muranova and W. Woess:
*Networks with complex weights: Green function and power series*, mdpi Mathematics 2022, 10, 820, 20pp. - W. Woess:
*Ratio limits and Martin boundary*, Documenta Math. 26 (2021) 1501-1528. - T. Hirschler and W. Woess:
*Laplace and bi-Laplace equations for directed networks and Markov chains*, Expositiones Mathematicae 39 (2021) 271-301. - E. Sava-Huss and W. Woess:
*Boundary behaviour of λ-polyharmonic functions on regular trees*, Annali di Matematica Pura ed Applicata 200 (2021) 35-50. Remark - M. Peigné and W. Woess:
*Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant*, Annals of Applied Probability 31 (2021) 2519-2537. - Ch. Lindorfer and W. Woess:
*The language of self-avoiding walks*, Combinatorica 40 (2020) 691-720. - T. Hirschler and W. Woess:
*Polyharmonic functions for finite graphs and Markov chains*, in "Frontiers in Analysis and Probability: in the Spirit of the Strasbourg-Zürich Meetings", pp. 77-90, Springer (2020). - M. A. Picardello and W. Woess:
*Multiple boundary representations of λ-harmonic functions on trees*, London Math. Soc. Lecture Notes 461 (2020) 95-125. - M. A. Picardello and W. Woess:
*Boundary representations of λ-harmonic and polyharmonic functions on trees*, Potential Analysis 51 (2019) 541-561. - J. Kloas and W. Woess:
*Multidimensional random walk with reflections*, Stochastic Proc. Appl. 129 (2019) 336-354. - A. Bendikov, W. Cygan and W. Woess:
*Oscillating heat kernels on ultrametric spaces*, Journal of Spectral Theory 9 (2019) 195-226. - T. Hirschler and W. Woess:
*Comparing entropy rates on finite and infinite rooted trees with length functions*, IEEE Trans. Information Theory 64 (2018) 5570-5580. - A. Bendikov, L. Saloff-Coste, M. Salvatori and W. Woess:
*Brownian motion on treebolic space: positive harmonic functions*, Ann. Institut Fourier (Grenoble) 66 (2016) 1691-1731. - T. Boiko and W. Woess:
*Moments of Riesz measures on Poincaré disk and homogeneous tree -- a comparative study*, Expositiones Math. 33 (2015) 353-374.

*ATTENTION: there is a stupid mistake in formula (2.8), relating the Euclidean with the hyperbolic Laplacian: a square is missing, the factor should be*(1-|z|^{2})^{2}/ 4 - J. Parkinson and W. Woess:
*Regular sequences and random walks in affine buildings*, Ann. Institut Fourier (Grenoble) 65 (2015) 675-707. - A. Bendikov, L. Saloff-Coste, M. Salvatori and W. Woess:
*Brownian motion on treebolic space: escape to infinity*, Revista Matematica Iberoamericana 31 (2015) 935-976. - A. Bendikov, A. Grigor'yan, Ch. Pittet and W. Woess:
*Isotropic Markov semigroups on ultra-metric spaces*, Uspekhi Mat. Nauk 69 (2014) No. 4 (418), 3-102. Link to*Russian version*. English original in Russian Math. Surveys 69 (2014) No. 4, 589-680. Remark 1 - Remark 2 - W. Woess:
*What is a horocyclic product, and how is it related to lamplighters ?*, Internat. Math. Nachrichten of the Austrian Math. Soc. 224 (2013) 1-27. Remark - W. Woess:
*On the duality between jump processes on ultrametric spaces and random walks on trees*, unpublished manuscript (2012). Remark - W. Woess:
*Context-free pairs of groups. II - Cuts, tree sets, and random walks*, Discrete Mathematics 312 (2012) 157-173. - T. Ceccherini-Silberstein and W. Woess:
*Context-free pairs of groups. I - Context-free pairs and graphs*, European J. Combinatorics 33 (2012) 1449-1466. - S. Brofferio, M. Salvatori and W. Woess:
*Brownian motion and harmonic functions on Sol(p,q)*, Internat. Math. Research Notes (IMRN) 22 (2012) 5182-5218. - M. Peigné and W. Woess:
*Stochastic dynamical systems with weak contractivity properties, II. Iteration of Lipschitz mappings*, Colloquium Math. 125 (2011) 55-81. - M. Peigné and W. Woess:
*Stochastic dynamical systems with weak contractivity properties, I. Strong and local contractivity. With a chapter featuring results of Martin Benda*, Colloquium Math. 125 (2011) 31-54. Remark - A. Bendikov, L. Saloff-Coste, M. Salvatori and W. Woess:
*The heat semigroup and Brownian motion on strip complexes*, Advances in Mathematics 226 (2011) 992-1055. - W. Huss, E. Sava and W. Woess:
*Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions*, Theoretical Computer Science 411 (2010), 3917-3922. - L. Saloff-Coste and W. Woess:
*Computations of spectral radii on G-spaces*, Contemporary Math. 484 (2009) 195-218. - F. Lehner, M. Neuhauser and W. Woess:
*On the spectrum of lamplighter groups and percolation clusters*, Mathematische Annalen 342 (2008) 69-89. - L. Bartholdi, M. Neuhauser and W. Woess:
*Horocyclic products of trees*, J. European Math. Society 10 (2008) 771-816. - D. I. Cartwright and W. Woess:
*The spectrum of the averaging operator on a network (metric graph)*, Illinois J. Math. 51 (2007) 805-830. - A. Karlsson and W. Woess:
*The Poisson boundary of lamplighter random walks on trees*, Geometriae Dedicata 124 (2007) 95-107. - R. Ortner and W. Woess:
*Non-backtracking random walks and cogrowth of graphs*, Canadian J. Math.59 (2007) 828-844. - M. Peigné and W. Woess:
*On recurrence of reflected random walk on the half-line*, with an appendix on results of Martin Benda, arXiv:math/0612306 (2006). Remark - S. Brofferio and W. Woess:
*Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs*, Potential Analysis 24 (2006) 245-265. - L. Saloff-Coste and W. Woess:
*Transition operators on co-compact G-spaces*, Revista Matematica Iberoamericana 22 (2006) 747-799. Remark - S. Brofferio and W. Woess:
*Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs*, Annales Inst. H. Poincaré (Prob. & Stat.) 41 (2005) 1101-1123,*Erratum*. - W. Woess:
*A note on the norms of transition operators on lamplighter graphs and groups*, Int. J. Algebra and Computation 15 (2005) 1261-1272. - L. Bartholdi and W. Woess:
*Spectral computations on lamplighter groups and Diestel-Leader graphs*, J. Fourier Analysis Appl. 11 (2005) 175 - 202. - W. Woess:
*Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions*, Combinatorics, Probability & Computing 14 (2005) 415-433. - D. I. Cartwright and W. Woess:
*Isotropic random walks in a building of type A*_{d}^{~}, Math. Zeitschrift 247 (2004) 101-135. - W. Woess:
*Generating function techniques for random walks on graphs*, in "Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces", P. Auscher, Th. Coulhon and A. Grigor'yan, eds. Contemporary Math. 338 (2003) 391-423. - T. Ceccherini-Silberstein and W. Woess:
*Growth-sensitivity of context-free languages*, Theoretical Comp. Sci. 307 (2003), 103-116. Remark - T. Ceccherini-Silberstein and W. Woess:
*Growth and ergodicity of context-free languages*, Transactions Amer. Math. Soc. 354 (2002) 4597-4625. - T. Nagnibeda and W. Woess:
*Random walks on trees with finitely many cone types*, J. Theoret. Probab. 15 (2002) 399-438. Remark - V. A. Kaimanovich and W. Woess:
*Boundary and entropy of space homogeneous Markov chains*, Ann. Probab. 30 (2002) 323-363. - W. Woess:
*Heat diffusion on homogeneous trees*, Bollettino Un. Mat. It. 4-B (2001) 703-709 and Erratum (caused by printer), Bollettino Un. Mat. It. 5-B (2002) 259-260. Remark 1, Remark 2 - S. Brofferio and W. Woess:
*On transience of card shuffling*, Proc. Amer. Math. Soc. 129 (2001) 1513-1519. - N. Seifter and W. Woess:
*Approximating graphs with polynomial growth*, Glasgow Math. J. 42 (2000) 1-8. - W. Woess:
*A note on tilings and strong isoperimetric inequality*, Math. Proc. Cambridge Phil. Soc.**124**(1998) 385-393. Remark - L. Saloff-Coste and W. Woess:
*Transition operators, groups, norms, and spectral radii*, Pacific J. Math.**180**(1997) 333-367. - P. J. Grabner and W. Woess:
*Functional iterations and periodic oscillations for random walk on the Sierpinski graph*, Stochastic Proc. Appl.**69**(1997) 127-138. - L. Saloff-Coste and W. Woess:
*Computing norms of group-invariant transition operators*, Combinatorics, Probability, and Computing**5**(1996) 161-178. - W. Woess:
*Dirichlet problem at infinity for harmonic functions on graphs*, International Conference on Potential Theory 1994, Proceedings (editors: J. Kral et al.), de Gruyter, Berlin (1996) 189-217. - V. A. Kaimanovich and W. Woess:
*Construction of discrete, non-unimodular hypergroups*, in: "Probability Measures on Groups and Related Structures", Proceedings, Oberwolfach 1994 (editor: H. Heyer), World Scientific, Singapore (1995) 196--209. Remark - W. Woess:
*The Martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree*, Monatshefte für Mathematik 120 (1995) 55-72. - W. Woess:
*Topological groups and recurrence of quasi transitive graphs*, Rendiconti Sem. Mat. Fis. Milano (1994) 185-213. - M. A. Picardello and W. Woess:
*The full Martin boundary of the bi-tree*, Ann. Probab. 22 (1994) 2203-2222. - D. I. Cartwright, V. A. Kaimanovich and W. Woess:
*Random walks on the affine group of local fields and of homogeneous trees*, Ann. Institut Fourier (Grenoble) 44 (1994) 1243-1288. - W. Woess:
*Random walks on infinite graphs and groups - a survey on selected topics*, Bull. London Math. Soc. 26 (1994) 1-60. Remark - S. Giulini and W. Woess:
*The Martin boundary of the Cartesian product of two hyperbolic spaces*, J. Reine Angew. Math. 444 (1993) 17-28. - W. Woess:
*Fixed sets and free subgroups of groups acting on metric spaces*, Math. Zeitschrift 214 (1993) 425-440. Remark - C. Thomassen and W. Woess:
*Vertex-transitive graphs and accessibility*, J. Combinatorial Th., Ser. B. 58 (1993) 248-268. - D. I. Cartwright, P. M. Soardi and W. Woess:
*Martin and end compactifications for nonlocally finite graphs*, Transactions Amer. Math. Soc. 338 (1993) 679-693. - W. Imrich, N. Sauer and W. Woess:
*The average size of nonsingular sets in a graph*, in: "Finite and Infinite Combinatorics in Sets and Logic", editor: N. Sauer et al., Kluwer, Dordrecht (1993), 199-205. - M. A. Picardello and W. Woess:
*Martin boundaries of Cartesian products of Markov chains*, Nagoya Math. J. 128 (1992) 153-169. - V.A. Kaimanovich and W. Woess:
*Behaviour at infinity and Dirichlet problem for random walks on graphs with a strong isoperimetric inequality*, Probability Theory Rel. Fields. 91 (1992) 445-466. - M. A. Picardello, M. H. Taibleson and W. Woess:
*Harmonic measure of the planar Cantor set from the viewpoint of graph theory*, Discrete Math. 109 (1992) 193-202. - D. I. Cartwright and W. Woess:
*Infinite graphs with nonconstant Dirichlet finite harmonic functions*, SIAM J. Discrete Math. 5 (1992) 380-385. - W. Woess:
*Behaviour at infinity and harmonic functions of random walks on graphs*, in "Probability Measures on Groups, X" (H. Heyer, editor), Proceedings, Oberwolfach (1990), Plenum Press, New York (1991) 437-458. - M. A. Picardello and W. Woess:
*Examples of stable Martin boundaries of Markov chains*, in "Potential Theory", (M. Kishi, editor), Proceedings, Nagoya 1990, de Gruyter, Berlin (1991) 262-270. - M. A. Picardello, M. H. Taibleson and W. Woess:
*Harmonic functions on Cartesian products of trees with finite graphs*, J. Functional Analysis 102 (1991) 379-400. - P. M. Soardi and W. Woess:
*Uniqueness of currents in infinite resistive networks*, Discrete Applied Math. 31 (1991) 37-49. - W. Woess:
*Topological groups and infinite graphs*, Discrete Math. 95 (1991) 373-384. Remark - P. M. Soardi and W. Woess:
*Amenability, unimodularity, and the spectral radius of random walks on infinite graphs*, Math. Zeitschrift 205 (1990) 471-486. - M. A. Picardello and W. Woess:
*Ends of infinite graphs, potential theory, and electrical networks*, in "Cycles and Rays: Basic Structures in Finite and Infinite Graphs", NATO ASI Series, Montréal. editor: G. Hahn, G. Sabidussi, R. E. Woodrow. Kluwer, Dordrecht (1990) 181-196. - W. Woess:
*Boundaries of random walks on graphs and groups with infinitely many ends*, Israel J. Math. 68 (1989) 271-301. - éW. Woess:
*Amenable group actions on infinite graphs*, Math. Annalen. 284 (1989) 251-265. - C. D. Godsil, W. Imrich, N. Seifter, M. E. Watkins and W. Woess:
*On bounded automorphisms of infinite graphs*, Graphs and Combinatorics 5 (1989) 333-338. - B. Mohar and W. Woess:
*A survey on spectra of infinite graphs*, Bull. London Math. Soc. 21 (1989) 209-234. - M. A. Picardello and W. Woess:
*A converse to the mean value property on homogenous trees*, Transactions Amer. Math. Soc. 311 (1989) 209-225. - W. Woess:
*Graphs and groups with tree-like properties*, J. Combinatorial Th., Ser. B, 47 (1989) 361-371. Remark - M. A. Picardello and W. Woess:
*Harmonic functions and ends of graphs*, Proc. Edinburgh Math. Soc. 31 (1988) 457-461. - M. A. Picardello and W. Woess:
*Finite truncations of random walks on trees*(appendix to: A. Korànyi, M. A. Picardello, M. Taibleson:*Hardy-spaces on non-homogeneous trees*), Symposia Math. 29 (1988) 255-265. (no download available) - W. Woess:
*Random walks on infinite graphs*, in: "Stochastics in Combinatorial Optimization", CISM Lecture Notes, Udine. editor: G. Andreatta, F. Mason, P. Serafini. World Scientific, Singapore (1987) 255-263. (no download available) - W. Woess:
*Context-free languages and random walks on groups*, Discrete Math. 67 (1987) 81-87. - M. A. Picardello and W. Woess:
*Martin boundaries of random walks: ends of trees and groups*, Transactions Amer. Math. Soc. 302 (1987) 185-205. - W. Woess:
*Harmonic functions on infinite graphs*, Rendiconti Sem. Mat. Fis. Milano 56 (1986) 51-63. - W. Woess:
*A description of the Martin boundary for nearest neighbour random walks on free products*, in "Probability Measures on Groups VIII" (editor: H. Heyer), Lecture Notes in Math. 1210(1986) 203-215. - W. Woess:
*Nearest neighbour random walks on free products of discrete groups*, Bollettino Unione Mat. Italiana 5-B (1986) 691-982. Remark - W. Woess:
*Transience and volumes of trees*, Archiv. d. Math. 46 (1986) 184-192. - W. Woess:
*A short computation of the norms of free convolution operators*, Proc. Amer. Math. Soc. 96 (1986) 167-170. Remark - P. Gerl and W. Woess:
*Local limits and harmonic functions for nonisotropic random walks on free groups*, Probability Theory Rel. Fields 71 (1986) 341-355. - P. Gerl and W. Woess:
*Simple random walks on trees*, European J. Combinatorics 7 (1986) 321-331. - M. A. Picardello and W. Woess:
*Random walks on amalgams*, Monatshefte für Mathematik 100 (1985) 21-33. - W. Woess:
*Random walks and periodic continued fractions*, Advances in Applied Probability 17 (1985) 67-84. - W. Woess:
*Chaotic random walks on certain abelian groups*, Colloquia Math. Soc. Janos Bolyai 36 (1985) 1147-1167. (no download available) - W. Woess:
*Harmonic functions on free groups*, in "Probabilités sur les Structures Géometriques", editor: G. Letac. Proceedings, Toulouse (1985) 141-153. - W. Woess:
*A random walk on free products of finite groups*, in "Probability Measures on Groups", Oberwolfach 1983, editor: H. Heyer. Springer Lect. Notes in Math. 1064 (1984) 467-470. - W. Woess:
*Cogrowth of groups and simple random walks*, Archiv d. Math. 41 (1983) 363-370. Remark - W. Woess:
*Puissances de convolution sur les groupes libres ayant un nombre quelconque de générateurs*, Inst. Elie Cartan 7 (1983) 181-190. (no download available) Remark - W. Woess:
*Périodicité de mesures de probabilité sur les groupes topologiques*, Inst. Elie Cartan 7 (1983) 170-180. Remark - W. Woess:
*A local limit theorem for random walks on certain discrete groups*, in "Probability Measures on Groups" (editor: H. Heyer), Lecture Notes in Math. 928 (1982) 467-477. Remark - W. Woess:
*Aperiodische Wahrscheinlickeitsmaße auf topologischen Gruppen*, Monatshefte für Mathematik 90 (1980) 339-345. - W. Woess:
*r-rekurrente Wahrscheinlichkeitsmaße auf topologischen Gruppen*, unpublished manuscript (1980). Remark

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Institut für Diskrete Mathematik

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