THE DIRICHLET PROBLEM ATINFINITY FOR RANDOM WALKSON GRAPHS WITH A STRONGISOPERIMETRIC INEQUALITYbyVadim A. Kaimanovichand Wolfgang WoessWith an extension toRiemannian manifoldsbyVadim A. KaimanovichOctober 1990Abstract. We study the spatial behaviour of random walks on in�nite graphs whichare not necessarily invariant under some transitive group action and whose transitionprobabilities may have in�nite range. We assume that the underlying graph G satis-�es a strong isoperimetric inequality and that the transition operator P is stronglyreversible, uniformly irreducible and satis�es a uniform �rst moment condition. Weprove that under these hypotheses the random walk converges almost surely to arandom end of G and that the Dirichlet problem for P -harmonic functions is solvablewith respect to the end compacti�cation. If in addition the graph as a metric space ishyperbolic in the sense of Gromov, then the same conclusions also hold for the hyper-bolic compacti�cation in the place of the end compacti�cation. The main tool is theexponential decay of the transition probabilities implied by the strong isoperimetricinequality. In an Appendix, it is shown how the same technique can be applied toBrownian motion to obtain analogous results for Riemannian manifolds satisfyingCheeger's isoperimetric inequality. In particular, in this general context new (andsimpler) proofs of well known results on the Dirichlet problem for negatively curvedmanifolds are obtained. 1. IntroductionLet G be an in�nite, connected, locally �nite graph. Consider a random walk(time-homogeneous Markov chain) Xn, n = 0; 1; 2; : : : , with state space G suchthat the one-step transition operator P is in some way adapted to the underlying1991 Mathematics Subject Classi�cation. 60J50; 31C12, 31C20, 58G32, 60J15.Key words and phrases. Transient random walk, harmonic functions, Dirichlet problem, spaceof ends, hyperbolic graph, Brownian motion.The �rst author was partially supported by Consiglio Nazionale delle Ricerche, GNAFATypeset by AMS-TEX1

2 VADIM A. KAIMANOVICH AND WOLFGANG WOESSgraph structure. Also, consider a compacti�cation G of G which is \natural" undera geometrical viewpoint. If (Xn) is transient, then the question arises whetherit converges almost surely in the topology of G to some random variable whichtakes its values in the boundary bG = G nG. In other words, we ask whether theboundary bG can serve as a model for the points attained at in�nity by our randomwalk. If the answer is positive, then for every starting point x 2 G we have thehitting distribution (harmonic measure) �x on bG . Assuming irreducibility of P ,all the harmonic measures �x are mutually absolutely continuous; we say that theyconstitute the harmonic measure class � on the boundary. Now, for every boundedmeasurable function h� on bG its Poisson integral h(x) = hh�; �xi is a boundedfunction on G harmonic with respect to the transition operator P of the chain.For di�erent functions h� 2 L1(bG; �) their Poisson integrals h are also di�erent,i.e. we have an imbedding of the space L1(bG; �) into the space H1(G;P ) of allbounded harmonic functions on G with the sup-norm. This imbedding is normpreserving.Now one can ask three di�erent questions about the boundary bG, the harmonicmeasures �x on it, and the corresponding Poisson integrals:(i) Does the Dirichlet problem admit solution: given a continuous functionon the boundary, does there exist a continuous extension to G which isharmonic on G? In other words, is it true that for every continuous functionh� its Poisson integral is continuous up to the boundary bG? One can easilysee that this is true if and only if for any sequence (xn) in G convergent toa point � 2 bG the corresponding harmonic measures �xn on bG convergeweakly to the point measure �� .(ii) Is the boundary bG large enough so that every bounded harmonic functionh on G can be obtained as the Poisson integral of some bounded measurablefunction h� on bG, i.e., is the imbedding L1(bG; �) ! H1(G;P ) an iso-morphism? If this is the case, then the boundary bG coincides as a measurespace with the Poisson boundary of the random walk.(iii) Is the compacti�cation G = G [ bG homeomorhic to the Martin compacti-�cation of the graph G corresponding to the transition operator P?The problem (ii) is a measure theoretic version of problem (iii), so that a positiveanswer to (iii) implies a positive answer to (ii). We emphasize that in generalthe Dirichlet problem for the Martin compacti�cation does not necessarily admitsolution (even in the case when the Martin boundary coincides with its active part,i.e. the support of the harmonic measure class is the whole Martin boundary -we give an example at the end of x6): a positive answer to (ii) or (iii) does notnecessarily imply a positive answer to (i). On the other hand the Dirichlet problemalways admits (trivial) solution for the one-point compacti�cation. Note also thatif the Dirichlet problem with respect to a nontrivial boundary bG admits solution,then the support of harmonic measure class coincides with bG, hence there existnonconstant bounded harmonic functions and the Poisson boundary of the randomwalk is nontrivial.In the present paper, we study convergence to the boundary and solvability ofthe Dirichlet problem 1) for the end compacti�cation (as introduced by Freudenthal[Fr]) for arbitraryG (x5) and 2) for the hyperbolic compacti�cation, if G - as a metric

THE DIRICHLET PROBLEM AT INFINITY 3space - is hyperbolic in the sense of Gromov [Gr] (x6).We prove convergence to the boundary and give positive answers to question(i) under the following assumptions: G satis�es a strong isoperimetric inequality,and the random walk is strongly reversible, uniformly irreducible and satis�es auniform �rst moment condition with respect to the distance in G; see x2 for detailsconcerning notation and preliminaries.We point out that no group invariance is assumed, and that we are particu-larly interested in a setting where (Xn) does not have bounded range (boundedrange means that one-step transitions can occur only at bounded distances). Inthis setting, our main tools are the exponential decay of the transition probabilities(a corollary of the strong isoperimetric inequality) and the uniform �rst momentcondition, which is a strengthening of tightness of the distance (step length) distri-butions. In x3 we show that under the latter assumption the distance increments of(Xn) can be controlled by i.i.d. random variables on the nonnegative integers whichhave �nite �rst moment. This in combination with the exponential decay of thetransition probabilities is used in x4 to derive distance estimates for (Xn) which areuniform with respect to the starting point. As a corollary, we obtain a weak formof the law of large numbers for the distance between Xn and a reference vertex.Observe that the uniform �rst moment condition gives a natural intermediate classbetween bounded range random walks and arbitrary walks. For example, it is the�niteness of the �rst moment that permits one to describe the Poisson boundary inintrinsic terms for a wide class of random walks on groups using a condition closeto the hyperbolicity, see Kaimanovich [K1].The main results, convergence to the boundary and the solution of the Dirichletproblem, are given in x5 for the end compacti�cation and in x6 for hyperbolicgraphs. At the end of x6, we give a brief review of results related to ours andpresent a simple example where the answer to question (i) is negative, even thoughthe harmonic measure class is supported by the whole of bG.One can also ask questions (i){(iii) for a Markov process with a continuousstate space equipped with a certain compacti�cation. For example, for Brownianmotion on negatively curved simply connected manifolds with the natural visibilitycompacti�cation, convergence to the visibility boundary sphere was proved by Prat[Pr], and the questions formulated above were answered positively by Anderson[An] (i), (ii) , Sullivan [Su] (i) , Anderson - Schoen [A-S] and Ancona [A1] (iii) ,respectively.In an Appendix, due to �rst author only, it is shown how the results obtainedhere for graphs can be reformulated for Brownian motion on Riemannian manifolds- the natural continous counterpart to reversible random walks on graphs. The mainassumption is Cheeger's isoperimetric inequality - indeed, the strong isoperimetricinequality for graphs was introduced by Dodziuk [Do] in analogy with Cheeger'sinequality. The latter is well known to be equivalent to the fact that �(M) (the topof the spectrum of Laplacian onM) is nonzero, i.e., to exponential decay of the heatkernel. This fact and boundedness of geometry alone are used to derive estimatesfor the rate of escape of the Brownian motion. Finally, convergence to the boundaryand solvability of the Dirichlet problem are obtained for the end and hyperboliccompacti�cations of Riemannian manifolds satisfying Cheeger's inequality.

4 VADIM A. KAIMANOVICH AND WOLFGANG WOESS2. Notation, preliminariesThroughout this paper, G denotes an in�nite, connected, locally �nite graph.Writing x 2 G we mean that x is a vertex of G. The edges are unoriented, loopsare permitted, but no multiple edges. By d(x; y) we denote the natural distance(number of edges in a shortest path) between the vertices x and y. We select, oncefor all, a reference vertex o and write jxj = d(x; o). The boundary of a set U � G is@U = fx 2 U j x has a neighbour in G n Ug :We shall always assume that G satis�es the following structure property.(IS) Strong isoperimetric inequality: there is a constant � > 0 such thatj@U j � � � jU j for every �nite U � G :See e.g. Dodziuk [Do], Gerl [Ge] and Ancona [A2] for properties of graphs satisfying(IS) .On G, we consider a random walk with Markov transition operator P =�p(x; y)�x;y2G , where p(x; y) is the probability of passing from x 2 G to y 2 Gin one step. As usual, we describe the position of the random walk at time n bythe nth projection Xn of the trajectory space � = GIN onto G ( IN is the set ofnonnegative integers): if � = (xn)n�0 2 � then Xn(�) = xn. On � , equipped withthe �-algebra induced by the Xn , we consider the family of probability measuresPrx, x 2 G, given by P and the initial point x. ThusPrx[X0 = x] = 1 ; Prx[X1 = y] = p(x; y) ;and Prx[Xn = y] = p(n)(x; y) ;the (x; y)-entry of the nth power Pn. The distance (step length) distribution atvertex x is given by�x(k) = Prx[d(X0; X1) = k] = Xy:d(x;y)=k p(x; y) ; k 2 IN :In order to study the (spacial) asymptotic behaviour of (Xn) on G, we need someassumptions which relate P with the structure of G.(SR) Strong reversibility (compare with [Ge]): P is reversible, i.e., there exists apositive measure (� function) m on G such thatm(x)p(x; y) = m(y)p(y; x) for all x; y 2 G ;and there are constants M1 > 0, M2 <1 such thatM1 � m(x) �M2 for every x 2 G :(UI) Uniform irreducibility (compare with Picardello and Woess [P-W] and An-cona [A2]): there are K 2 IN and c > 0 such that d(x; y) = 1 impliesp(k)(x; y) � c for some k � K.(UM) Uniform �rst moment condition: we assume1Xn=1�(n) <1 ; where �(n) = supx2G�x�[n;1)� :

THE DIRICHLET PROBLEM AT INFINITY 5Note that for an irreducible random walk on (a Cayley graph of) a �nitelygenerated group, (UI) is automatically true, while (UM) becomes �niteness of the�rst moment of the underlying probability measure on the group. If the measureon the group is symmetric, then the random walk is strongly reversible with respectto the Haar (� counting) measure on the group. In our general setting, it is easyto see that uniform irreducibility implies that G has bounded vertex degrees:jfy j d(x; y) = 1gj � K=c for every x 2 G :Hence there exists a constant D > 0 such that(2.1) jfy j d(x; y) � ngj � Dn for every x 2 G and n � 0 :The transition operator P acts on functions h : G! IR byPh(x) =Xy p(x; y)h(y) ;whenever this series converges for all x 2 G. A harmonic function is a functionsatisfying Ph = h.We shall frequently use the following generalization of the results of [Ge] torandom walks with in�nite range.Theorem A (Kaimanovich [K2]). Under conditions (SR) and (UI), the strongisoperimetric inequality implies that kPk < 1, where kPk is the norm of P as anoperator on L2(G;m) , and there is a constant M > 0 such thatp(n)(x; y) �M kPkn for all x; y 2 G; n � 0 :3. The uniform first moment conditionand polymorphisms of the trajectory spaceIn this section we study in detail the meaning of the uniform �rst momentcondition (UM).De�nition 1. A family (�i)i2I of probability distributions on IN is called tight(see e.g. Billingsley [Bi]) if�(n) = supi2I �i�[n;1)�! 0 as n!1 ;and it satis�es the uniform �rst moment condition if in addition1Xn=1�(n) <1 :

6 VADIM A. KAIMANOVICH AND WOLFGANG WOESSDe�nition 2. Given two probability distributions � and � on IN , we say that �dominates � (notation � � �) if��[n;1)� � ��[n;1)� for every n � 0 :Equivalently, this says that F� � F� for the corresponding distribution functions.One can easily see that this property means that � can be obtained from � byredistributing the mass of � from larger values n 2 IN to smaller ones, i.e. � � �i� there exists a family (�n)n2I of probability measures on IN such that supp �n �[0; n] and � =Pn �(n) �n .Lemma 1. A family (�i)i2I of probability distributions on IN is tight if and onlyif there exists a probability distribution � on IN which dominates all of the �i . Inaddition, (�i)i2I satis�es the uniform �rst moment condition if and only if � canbe chosen to have a �nite �rst moment.Proof. If �i � � for all i 2 I then�(n) = supi2I �i�[n;1)� � ��[n;1)�! 0 as n!1 :If in addition � has a �nite �rst moment then1Xn=1�(n) � 1Xn=1��[n;1)� = 1Xk=1 k �(k) <1 :Conversely, assume tightness and let �(n) be as in De�nition 1. Then �(0) = 1, and�(n) � 0 is decreasing. Thus, setting �(n) = �(n)� �(n+ 1) de�nes a probabilitydistribution on IN , and � � �i for all i 2 I. The �rst moment of � is1Xk=1 k �(k) = 1Xn=1�(n) :De�nition 3 (Vershik [Ve]). A polymorphism of two probability spaces (Y;Pr) and(Y 0;Pr0) is a triple consisting of a probability space (�; IP ) and two measurable,measure preserving mappings� : �! Y and �0 : �! Y 0 :We illustrate the situation with the following diagram:(Y;Pr) � �(�; IP ) �0�!(Y 0;Pr0) :Thus (�; IP ) subsumes the information contained in both of the given probabilityspaces. (In our notation, we omit the respective �-algebras.)Let � be a probability distribution on IN , the set of nonnegative integers. By(IN1; �1) we denote the product probability space consisting of all sequences(kn)n�1 in IN equipped with the usual �-algebra generated by the projectionsZn : IN1 ! IN and carrying the in�nite product measure arising from �. Inparticular, the Zn , n � 1, are i.i.d. with distribution �.

THE DIRICHLET PROBLEM AT INFINITY 7Proposition 1. The distance distributions �x , x 2 G, of the random walk onG given by P are tight if and only if there exist a probability space (�; IP ) and aprobability distribution � on IN with the following property: for every x 2 G thereexists a polymorphism (�;Prx) �x �(�; IP ) ��!(IN1; �1)such that for IP -almost every � 2 �d(Xxn�1; Xxn) � Z�n ; n = 1; 2; 3; : : : ;where Xxn(�) = Xn��x(�)� and Z�n (�) = Zn��(�)�. In addition, the random walksatis�es the uniform �rst moment condition if and only if � can be chosen to havea �nite �rst moment.Proof. Suppose that the distance distributions �x , x 2 G, are tight. By Lemma 1there exists a probability distribution � on IN which dominates all of the �x . Weequip IN1 with the corresponding product measure �1.In order to construct the required polymorphism, we de�ne � = [0; 1)1, thespace of all sequences � = (�n)n�1 in [0; 1) equipped with the usual Borel product�-algebra. For IP we choose the in�nite product of the Lebesgue measure on [0; 1).Thus, the projections of � onto [0; 1) are independent, uniformly distributed randomvariables.We subdivide [0; 1) into consecutive intervalsI0 = [0 = a0; a1) ; I1 = [a1; a2) ; : : : ;of lengths ai+1 � ai = �(i) ; i � 0 :If � = (�n)n�1 2 � then we de�ne �(�) = (kn)n�1, where(3.1) kn = i () �n 2 Ii ; i � 0 :Thus, � : (�; IP )! (IN1; �1) is measurable and measure preserving.Now choose x 2 G and choose an enumeration, depending on X, of the verticesof G: G = fx` j ` 2 INg, in such a way that x0 = x and `0 � ` implies d(x`0 ; x) �d(x`; x) . In analogy with the above construction, subdivide [0; 1) into successiveintervals, denoted Jx;x` , of length p(x; x`), ` � 0. Do this for every x 2 G. Thus,for n � 0 the set Sy:d(x;y)�n Jx;y is an interval [0; bx;n) of length bx;n = �x([0; n]) ��([0; n]) , i.e. [y:d(x;y)�n Jx;y � n[i=0 Ii :In other words,(3.2) d(x; y) � n implies Jx;y � 1[i=n Ii :

8 VADIM A. KAIMANOVICH AND WOLFGANG WOESSIf � = (�n)n�1 2 � , then we de�ne �x(�) = � = (xn)n�0, where(3.3) x0 = x and xn = y () �n 2 Jxn�1;y ; n � 1 :Thus, �x : (�; IP )! (�;Prx) is measurable and measure preserving.Finally, let � = (�n)n�1 2 � , (xn)n�0 = �x(�) and (kn)n�1 = �(�) . Ifd(xn�1; xn) = ` then Jxn�1;xn � S1i=` Ii by (3.2). By (3.3) we have �n 2 Jxn�1;xn .Hence (3.1) yields kn � `. Thusd(Xxn�1; Xxn) � Z�nas proposed.Conversely, it is obvious that the existence of a polymorphism with the statedproperties yields �x � � for every x 2 G.The above proposition allows us to majorize the distance increments of (Xn)with i.i.d. random variables Zn on IN independently of the starting point x 2 G.4. Distance estimatesIn this Section we derive several estimates for the random walk on G resultingfrom the assumptions (IS), (SR), (UI), (UM) made in x2, which will be used belowfor studying convergence to the boundary and the Dirichlet problem.Lemma 2. Suppose that the random walk on G satis�es (UM). Then we have thefollowing:(a) limn!1 1nd(Xn�1; Xn) = 0Prx-almost surely for every x 2 G, and convergence in probability is uniformin x : limn!1Prx[supk�n 1kd(Xk�1; Xk) > "] = 0uniformly in x 2 G for every " > 0;(b) limn!1Prx[ 1n supk�n d(Xk�1; Xk) > "] = 0uniformly in x 2 G for every " > 0;(c) lim supn!1 1n supk�n d(X0; Xk) � �� = 1Xn=1�(n)Prx-almost surely for every x 2 G, andlimn!1Prx[ 1n supk�n d(X0; Xk) � C1] = 0uniformly in x 2 G for every C1 > ��.

THE DIRICHLET PROBLEM AT INFINITY 9Proof. Consider the polymorphism constructed in Proposition 1. Then(4.1) d(Xxn�1; Xxn) � Z�n IP -a.s. for every x 2 G :By construction, the Z�n are i.i.d. with common distribution � having �nite �rstmoment ��. By the law of large numbers,(4.2) limn!1 1n nXk=1Z�k = �� IP -a.s.Hence, limn!1 1nZ�n = 0 IP -a.s.This and (4.1) together with the fact that �x is measure preserving yield (a).Relation (4.1) also yields1n supk�n d(Xx0 ; Xxk ) � 1n nXk=1Z�k IP -a.s.In combination with (4.2), we obtain (c). Finally,IP [ 1n supk�n d(Xxk�1; Xxk ) > "] � nXk=1 IP [Z�k > "n] � n�(" n) :As � is decreasing and P1n=1 �(n) < 1 , we have limn!1 n�(" n) = 0 for every" > 0. This yields (b).Lemma 3. Suppose that the graph G and the random walk satisfy assumptions(IS), (SR) and (UI). Then there exists a constant C0 > 0 such thatlim infn!1 1n infk�n d(X0; Xk) > C0Prx-almost surely for every x 2 G, andlimn!1Prx[ infk�n 1n d(X0; Xk) � C0] = 0uniformly in x 2 G.Proof. Let An = [ infk�n d(X0; Xk) � C0n] :From Theorem A and (2.1) we getPrx(An) � 1Xk=nPrx[d(X0; Xk) � C0n] = 1Xk=n Xy:d(x;y)�C0n p(k)(x; y)� 1Xk=nM � kPkk � jfy j d(x; y) � C0ngj � M1� kPk �DC0kPk�n :Thus, if we choose C0 > 0 small enough such thatDC0kPk < 1, thenP1n=1 Prx(An)converges uniformly in x 2 G. Now the Borel { Cantelli Lemma yields the result.

10 VADIM A. KAIMANOVICH AND WOLFGANG WOESSCorollary 1. Under assumptions (IS), (SR), (UI) and (UM)0 < C0 < lim infn!1 1n jXnj � lim supn!1 1n jXnj < C1 <1Prx-almost surely for every x 2 G.This is a \law of large numbers" with respect to the distance in G. For anirreducible random walk with �nite �rst moment on a �nitely generated group theconditions (UI) and (UM) are satis�ed with respect to the Cayley graph (� word)metric(s) of the group. In this case, jXnj=n converges almost surely to a constant(the rate of escape) by Kingman's subadditive ergodic theorem (see e.g. Derriennic[D2]). This limit is always positive if the group is nonamenable, i.e. the Cayleygraphs of the group satisfy (IS), see Kaimanovich and Vershik [K-V]. Corollary 1generalizes these facts (in a weaker form) to a non group-invariant setting. Comparealso with Varopoulos [Va].Lemma 4. Suppose that (IS), (SR), (UI) and (UM) hold. If " > 0 is su�cientlysmall, then limjxj!1Prx[ jXnj � " jxj for some n � 0] = 0 :Proof. Let " < 1=2 and � > 0. ThenPrx[ 9n : jXnj � " jxj ] �� Prx[ 9n � � jxj : jXnj � " jxj ] + Prx[ 9n � � jxj : jXnj � " jxj ] ;wherePrx[ 9n � � jxj : jXnj � " jxj ] � Prx[ 9n � � jxj : d(X0; Xn) � (1� ")jxj ]= Prx[ 1� jxj supn��jxj d(X0; Xn) � (1� ")=� ]� Prx[ 1� jxj supn��jxj d(X0; Xn) � 1=2� ]and Prx[ 9n � � jxj : jXnj � " jxj ] � 1Xn=�jxj Xy:jyj�"jxj p(n)(x; y)� M1� kPk (kPk�D")jxjas follows from Theorem A and (2.1). Now set � = 1=2C1 and choose " > 0 smallenough such that kPk�D" < 1. Then applying Lemma 2(c) we get the desiredresult.Combining Lemmas 2{4, we obtain the following technical corollary, which willbe useful in the sequel.

THE DIRICHLET PROBLEM AT INFINITY 11Corollary 2. Suppose that (IS), (SR), (UI) and (UM) hold. Let C0 be as inLemma 3, and let �; " > 0. For x 2 G, consider the event Ax = Ax(�; ") in thetrajectory space de�ned by the following �ve properties:(i) X0 = x,(ii) d(Xn; Xn+1) � " jxj for all n � � jxj,(iii) d(Xn; Xn+1) � " n for all n � � jxj,(iv) d(X0; Xn) > C0 n for all n � � jxj,(v) jXnj > " jxj for all n � 0 .There exists a constant "0 > 0 (independent of �) such that if " � "0 , thenlimjxj!1Prx(Ax) = 1 :5. Convergence to the boundary and Dirichlet problemin the end compactificationWe recall the construction of the end compacti�cation of a graph G, see Freu-denthal [Fr], Jung [Ju], Cartwright, Soardi and Woess [C-S-W].An in�nite path is a one-sided in�nite sequence p of successively adjacent verticeswithout repetitions. Two in�nite paths p, p0 are equivalent if for every �nite U � G,all but �nitely many of the vertices of p and p0 lie in the same connected componentof G n U . An end is an equivalence class under this relation. The set of all ends isdenoted by . If U � G is �nite, then we add to each component of G nU all endshaving a representative path lying entirely in that component. If z 2 G[nU , thenamong the components augmented in this way there is precisely one which containsz. This is the component of z with respect to U , denoted by C(U; z). Varying U(�nite) and z, the family of all sets C(U; z) becomes a basis of the end topology. Inthis way, G[ becomes a compact, totally disconnected Hausdor� space in whichG (� the vertex set of G) is discrete, open and dense.We remark that the space of ends can also be introduced as a projective limit:let U be a �nite (� compact) subset of G. Denote by CU the (�nite, discrete) spaceof all in�nite connected components of GnU . For any two sets U1 � U2 there existsa natural projection CU2 ! CU1 . The projective limit of the system fCU j U � Ggis another model for . See e.g. [C-S-W] for more details.We begin with the following simple statement.Lemma 5. Every sequence (xn) of vertices in G such thatjxnj+ jxn+1j � d(xn; xn+1)!1converges to an end.Proof. Take a �nite subset U � G. Suppose two points x; y 2 G n U belong todi�erent connected components of the set G nU . It means that all the paths (and,in particular, the shortest paths) connecting x and y pass through the set U . Hencethere exists a point z 2 U such thatd(x; y) = d(x; z) + d(z; y) � jxj+ jyj � 2jzj � jxj+ jyj � 2RU ;

12 VADIM A. KAIMANOVICH AND WOLFGANG WOESSwhere RU = maxz2U jzj :Thus the condition of the Lemma implies that for every �nite U there existsan integer N and a set C(U; !) such that xn 2 C(U; !) for all n � N . Moreover,xn ! 1. Since the sets C(U; !) form a basis of the end topology, the Lemma isproven.Theorem 1. Assume that the graph G and the random walk satisfy assumptions(IS), (SR), (UI) and (UM). Then there exists an -valued random variable X1such that for every x 2 G,limn!1Xn = X1 Prx-almost surelyin the end topology.Proof. Choose C0 as in Lemma 3. Consider the event E in the trajectory spacede�ned by(5.1) limn!1 1n d(Xn; Xn+1) = 0 and lim infn!1 1n d(X0; Xn) > C0 :Then Prx(E) = 1 by Lemmas 2(a) and 3. But as follows from Lemma 5, every(Xn) 2 E converges to an end of G.In view of Theorem 1, we may de�ne the family of hitting distributions (harmonicmeasures) �x, x 2 G, on : for a Borel set B � ,(5.2) �x(B) = Prx[X1 2 B] :Observe that for every continuous function h� on , the Poisson integral(5.3) h(x) = Z h� d�xde�nes a bounded harmonic function on G. We are now ready to formulate thesolution of the Dirichlet problem with respect to the end compacti�cation.Theorem 2. Under the assumptions of Theorem 1, every continuous function h�on admits a unique continuous extension to G[ which is harmonic on G. Theextension is given by (5.3).Proof. Uniqueness follows from the maximum principle. In view of (5.2) and (5.3),what we have to show is that limx!! �x = �! weakly,whenever x converges in G to ! 2 (here �! denotes the unit mass at !). It issu�cient to show that limx!! �x�C(U; !) \ � = 1

THE DIRICHLET PROBLEM AT INFINITY 13for every basic neighbourhood C(U; !) of !, where U � G is �nite.To prove this, consider the set Ax � � de�ned in Corollary 2 (we shall specifythe values of � and " later) and take a trajectory � 2 Ax . Then for Xn = Xn(�)jXnj+ jXn+1j � d(Xn; Xn+1) > " jxjfor all n � � jxj , andjXnj+ jXn+1j � d(Xn; Xn+1) > 2 (C0 n� jxj)� " n= (2C0 � ")n� 2 jxj� �(2C0 � ")�� 2� jxjfor all n � � jxj . Take now " < min("0; 2C0) and � = ("0 + 2)=(2C0 � "0) with "0as in Corollary 2. Thus we have thatjXnj+ jXn+1j � d(Xn; Xn+1) > "0 jxjfor all n � 0 .Now from the proof of Lemma 5 follows that if x 2 C(U; !) and " jxj > 2RU ,then none of the trajectories in the set Ax ever leaves the set C(U; !) . Hence�x�C(U; !) \ � � Prx(Ax)! 1 ; as x! ! :6. Convergence to the boundary and Dirichlet problemon hyperbolic graphsIn this section we assume that G, viewed as a metric space with the naturaldiscrete metric d , is a hyperbolic space in the sense of [Gr]; see also Ghys and de laHarpe [G-H], Short [Sh] or Coornaert, Delzant, Papadopoulos [C-D-P]. We reviewthe basic features and point out the necessary adaptations.With respect to the reference vertex o, de�ne for x; y 2 G the Gromov product(xjy) = 12�jxj+ jyj � d(x; y)� :The graph G is hyperbolic if there exists a � > 0 (possibly large) such that for allx; y; z 2 G(6.1) (xjz) � minf(xjy); (yjz)g� � :On the basis of [G-H, x7.2], we describe the hyperbolic compacti�cation. Assumethat G is hyperbolic. (Observe that our de�nition of hyperbolicity involves onlyone base point, so that the � of [G-H] has to be chosen equal to twice times the �of (6.1), compare with [Gr].) Choose a > 0 such that a0 = e2�a � 1 < p2 � 1 andde�ne for x; y 2 G(6.2) �a(x; y) = � 0; if x=y;e�a(xjy); otherwise.

14 VADIM A. KAIMANOVICH AND WOLFGANG WOESSThen, precisely as in [G-H, x7.2], one veri�es that �a is symmetric, �a(x; y) = 0 ifand only if x = y, and(6.3) �a(x0; xn) � 11� 2a0 nXi=0 �a(xi�1; xi)for all n > 0 , x0; : : : ; xn 2 G. Thus, �a is \almost" a metric. If we de�ne forx; y 2 G�a(x; y) = inff nXi=1 �a(xi�1; xi) j n � 1; x = x0; x1; : : : ; xn = y 2 Gg ;then �a is a metric on G, and(6.4) (1� 2a0)�a(x; y) � �a(x; y) � �a(x; y) for all x; y 2 G :This can be proved exactly as in [G-H, x7.2]. The completion bG of G with respect to�a as a topological space does not depend on the choice of a. The induced topologyon bG is the hyperbolic topology. In this way, bG becomes a compact Hausdor� spacein which G is discrete, open and dense. The hyperbolic boundary is @G = bG nG.Thus, a sequence (xn) of vertices of G converges to some hyperbolic boundarypoint if and only if limm;n!1(xmjxn) =1 ;and two such sequences (xn) and (yn) converge to the same boundary point if andonly if limm;n!1(xmjyn) =1 :(Because of (6.1) this is an equivalence relation, and the boundary can be introducedas the set of equivalence classes.)We remark that it is easy to verify (cf. Lemma 5) that for hyperbolic G, thehyperbolic compacti�cation is �ner that the end compacti�cation: the identity onG extends to a continuous surjection bG! G [ , which maps @G onto . On theother hand, even a graph with a rich end structure need not to be hyperbolic ingeneral.With an abuse of the notation introduced in x5, we shall write X1 and �x,x 2 G, for the limit random variable of (Xn) and their Prx-distributions on @G,whose existence is guaranteed by the following theorem.Theorem 3. Assume that G is a hyperbolic graph satisfying the condition (IS), andthat P gives rise to a random walk on G with the properties (SR), (UI) and (UM).Then there exists a @G-valued random variable X1 such that for every x 2 Glimn!1Xn = X1 Prx-almost surelyin the hyperbolic topology.Proof. Consider the set E of trajectories de�ned by (5.1) in the proof of Theorem1 . Then Prx(E) = 1 , and if � 2 E then for Xn = Xn(�)lim infn!1 1n (XnjXn+1) > C0 :

THE DIRICHLET PROBLEM AT INFINITY 15Consequently, for m � n we have by (6.3)�a(Xn; Xm) � 11� 2a0 1Xk=n e�a(XkjXk+1) ;which tends to zero as n!1. Therefore (Xn) is a Cauchy sequence and convergesin the hyperbolic topology. As jXnj ! 1 , the limit must be a boundary point.We de�ne the hitting probabilities of (Xn) on @G as mentioned above: if x 2 Gand B is a Borel set in @G, then(6.5) �x(B) = Prx[X1 2 B] :Once more, the Dirichlet problem admits solution.Theorem 4. Under the assumptions of Theorem 3, every continuous function h�on @G has a unique continuous extension to bG which is harmonic on G. Theextension is given by the Poisson integralh(x) = Z@G h� d�x ; x 2 G :Proof. We proceed precisely as in the proof of Theorem 2. Let � 2 @G and considera basic neighbourhoodN(�; s) = f� 2 bG j �a(�; �) < sg ; s > 0 :In order to show that limx2G;x!� �x�n(�; s) \ @G� = 1 ;we consider the set Ax � � de�ned in Corollary 2 and choose(6.6) " < minf"o; C0=2g ; � = 2=CO :Let � 2 Ax , Xn = Xn(�). We claim that if jxj is large enough then �a(x;Xn) <s=2 for all n. Thus, if in addition x 2 N(�; s=2) , then the limit points of all thetrajectories in Ax lie in N(�; s) , and�x�N(�; s) \ @G� � Prx(Ax)! 1 ; as x! � :To prove the claim, observe that for n � 2C0 jxj,jXnj > C0 n� jxj � C02 nby (iv) of Corollary 2, so that in combination with (iii) and (6.6) we get(XnjXn+1) > 12(C0 n� " n) > C04 n :

16 VADIM A. KAIMANOVICH AND WOLFGANG WOESSOn the other hand, for n � 2C0 jxj,(XnjXn+1) � 12(2" jxj � " jxj) = "2 jxjby (v) and (ii). Now (6.3) yields(1� 2a0)�a(x;Xn) � 1Xk=0 e�a(XkjXk+1)� 2C0 jxj e�a"jxj=2 + 1Xk=2jxj=C0 e�aC0k=4 = �(jxj) ;which tends to zero as jxj ! 1 . If jxj is large enough such that �(jxj) < (1 �2a0)s=2 , then by (6.4) we have �a(x;Xn) < s=2 for all n, as proposed.Before concluding, we remark that in all results of this paper, (IS) and (SR) canbe replaced by the following weaker hypothesis:(E) There are � < 1 and M > 0 such thatp(n)(x; y) �M �n for all x; y 2 G; n � 0 :We now brie y review results related to ours concerning random walks on graphsand groups.If (Xn) has �nite range (i.e., supfd(x; y) j p(x; y) > 0g < 1) and is transient,then it is easy to verify that it converges a.s. to a random end, without furtherhypotheses. For nearest neighbour random walks (i.e., p(x; y) > 0 () x and y areneighbours), the Dirichlet problem with respect to the end compacti�cation admitssolution if and only if the Green kernel vanishes at in�nity, given that there areat least two ends, see Benjamini and Peres [B-P] for trees and Cartwright, Soardiand Woess [C-S-W] for even non locally �nite graphs. The (similar) proofs in [B-P]and [C-S-W] carry over immediately to �nite range random walks on locally �nitegraphs.In a group-invariant setting, the solution of the Dirichlet problem has been givenmuch earlier by Derriennic [D1] for �nite range random walks on �nitely generatedfree groups (� homogeneous trees) as a by-product of his identi�cation of the spaceof in�nite words (� space of ends) with the Martin boundary. Convergence to arandom end for group invariant random walks on trees has been proved by Sawyerand Steger [S-S] assuming a moment condition, see also Sawyer [Sa] and Cartwrightand Sawyer [C-Sa].An earlier approach due to Furstenberg turned out to be very useful in provingthat a.e. random walk trajectory converges to a (random) boundary point withoutany additional conditions imposed on the walk. The idea consists in verifying thatthe group action on the boundary satis�es a certain convergence property. It was�rst applied for proving the convergence of random walks on Fuchsian groups to theboundary circle of the Poincar�e disc [Fu, Thm.1.3]. Margulis remarked that thismethod also works for the end compacti�cation of �nitely generated free groups

THE DIRICHLET PROBLEM AT INFINITY 17(unpublished, cf. [K-V]). Independently, Cartwright and Soardi [C-So] used thisapproach for proving convergence to a random end for group invariant random walkson trees in general; a special case (amenability of the group) has to be excludedhere. For irreducible random walks which are invariant under a nonamenable,vertex-transitive group acting on a graph with more than two ( () in�nitelymany) ends, the same method yields convergence to a random end and solvabilityof the Dirichlet problem without any moment condition, see Woess [Wo]. Thisapplies in particular to all �nitely generated groups with in�nitely many ends. Thesame technique easily permits one to prove convergence to a random boundarypoint and solvability of the Dirichlet problem for the hyperbolic compacti�cationof nonamenable hyperbolic groups as well.For general (non group-invariant) hyperbolic graphs, the only known result is dueto Ancona [A2]: it applies to bounded range random walks satisfying (UI) and suchthat lim sup p(n)(x; y)1=n < 1 for some (() all) x; y 2 G. In this case, the Martinboundary is identi�ed with the hyperbolic boundary, so that a.s. convergence tothe hyperbolic boundary is veri�ed; solvability of the Dirichlet problem is provedunder the assumption that the Green kernel vanishes at in�nity. This generalizesearlier results of Series [Se] for Fuchsian groups.We now give the example, announced in the introduction, of a random walkon a graph (a tree), where the whole Martin boundary is active, but the Dirichletproblem does not admit solution.Example. Let T0 be the in�nite binary tree with root r0 : every vertex has threeneighbours except the root, which has only two. Construct a tree Tn from T0 byconnecting a �nite path of length n3 to r0. As the root of Tn, take the vertex rn ofTn which has only one neighbour. Now build a tree T from all the Tn by drawingan edge between rn and rn+1 , n � 0 : see the �gure.Being a tree, T is a hyperbolic graph, and its end compacti�cation coincideswith the hyperbolic one. The space of ends of T is given by the union of the spacesof ends of all the Tn plus an additional one, denoted !1, which is the limit of (rn).Because of the presence of �nite unrami�ed paths (successive vertices with onlytwo neighbours) of arbitrary length, T does not satisfy the strong isoperimetricinequality.Let P be the transition operator of the simple random walk on T , which movesfrom a vertex of T to any of its neighbours with equal probability. It is easily seento be transient.By Cartier [Ca], the Martin compacti�cation of (T; P ) coincides with the endcompacti�cation.

18 VADIM A. KAIMANOVICH AND WOLFGANG WOESSFigureLet g(x; y) be the Green kernel of P , i.e. g(x; y) = P1n=0 p(n)(x; y). By [B-P]and [C-S-W], an end ! of T is regular for the Dirichlet problem (i.e., �x ! �!weakly as x! !) if and only if(6.7) limx!! g(x; r0) = 0 :Using the well known behaviour of the simple random walk on the binary tree, oneobtains that (6.7) holds for every ! 2 n f!1g . On the other hand, using forexample the methods of Gerl and Woess [G-W], one can prove thatlimn!1 g(rn; r0) > 0(we omit the details of the calculation).Thus, the Dirichlet problem for P -harmonic functions is not solvable with respectto . On the other hand, being regular, all points of n f!1g lie in the supportof the harmonic measure class �. But the closure of the latter set is , so that thesupport of � is the whole boundary.Appendix: Convergence to the boundary and Dirichlet problemfor Brownian motion onRiemannian manifoldsbyV. A. KaimanovichIn this Section we show how the results obtained above can be reformulated forBrownian motion on Riemannian manifolds.Let M be a complete connected Riemannian manifold with the Riemannianmetric d . Fix a reference point x0 2M and denote by jxj the distance d(x; x0) forall x 2M . We assume that M has bounded geometry , i.e. the sectional curvatureson M are uniformly bounded and the injectivity radius is bounded away from zero.The Laplace - Beltrami operator � of the Riemannian metric on M determines adi�usion process with state space M and generator � which is called the Brownianmotion on M . Denote by Prx the probability measure in the space of sample paths(Xt)t�0 of the Brownian motion starting from a point x 2 M , and by pt(x; �)the density of the one-dimensional distribution of the measure Prx at time t withrespect to the Riemannian volume m , i.e. the fundamental solution of the heatequation on M (see e.g. Ikeda and Watanabe [I-W] or Pinski [Pi]).Now consider the Riemannian analogues of the discrete conditions introduced inx2.It is well known that the densities pt are symmetric: pt(x; y) = pt(y; x) for allx; y 2 M . In other words, the Brownian motion is reversible with respect to theRiemannian volume. This corresponds to the fact that the Laplace - Beltramioperator � is self-adjoint in the space L2(M;m) . It is the natural analogue ofcondition (SR).

THE DIRICHLET PROBLEM AT INFINITY 19Boundedness of the geometry of M implies that the growth of the volume ofRiemannian balls in M is uniformly bounded (see Cheeger and Ebin [C-E]): thereexists a constant D > 0 such thatmfy 2M : d(x; y) � rg � Dr for every x 2M and r � 0 :This is an analogue of the inequality (2.1).From the comparison theorems for Brownian motion and standard scaling esti-mates for the decay at in�nity of the heat kernel follows [Pi] that there exists ameasure � on IR with a �nite �rst moment such thatPrx[ sup0���1 d(X0; X�) � r] � �([r;1)) for all x 2M ; r � 0 ;which is an analogue of condition (UM) (in fact, this measure � can be taken tohave Gaussian decay).Finally, instead of conditions (IS) and (UI) we can directly assume that 0 doesnot belong to the spectrum of the Laplace - Beltrami operator � in the spaceL2(M;m) , i.e. �(M) < 0 for the top of the spectrum of �. This condition is wellknown to be equivalent to Cheeger's isoperimetric inequality :area(@U) � � �m(U)for a certain constant � = �(M) > 0 and for every compact connected subdomainU � M with smooth boundary @U ; see Cheeger [Che] and Buser [Bu]. It impliesthat there exist constants M > 0 ; � < 1 such thatpt(x; y) �M�t for all x; y 2M ; t � 0 :Now, reproducing the argument from x4 we get the following statements.Lemma 6. Let M be a complete connected Riemannian manifold with boundedgeometry such that �(M) 6= 0. Then there exist constants C0; C1 such that0 < C0 < lim inft!1 1t jXtj � lim supt!1 1t jXtj < C1 <1and limt!1 1t sup0���1 d(Xt; Xt+� )! 0Prx-almost surely for every x 2M .Lemma 7. Let M be a complete connected Riemannian manifold with boundedgeometry such that �(M) < 0. For x 2 M , consider the event Ax = Ax(�; ")in the trajectory space of the Brownian motion on M de�ned by the following �veproperties:(i) X0 = x,(ii) d(Xt; Xt+� ) � " jxj for all t � � jxj ; 0 � � � 1,(iii) d(Xt; Xt+� ) � " t for all t � � jxj ; 0 � � � 1,(iv) d(X0; Xt) > C0 t for all t � � jxj,(v) jXtj > " jxj for all t � 0 .

20 VADIM A. KAIMANOVICH AND WOLFGANG WOESSThere exists a constant "0 > 0 (independent of �) such that if " � "0 , thenlimjxj!1Prx(Ax) = 1 :Now, proceeding as in xx5,6 we get the following analogue of Theorems 1{4.Theorem 5. Let M be a complete connected Riemannian manifold with boundedgeometry such that �(M) 6= 0. Then there exists a random variable X1 takingvalues in the space of ends (M) of the manifold M such that for every x 2M ,limt!1Xt = X1 Prx-almost surelyin the end topology. The Dirichlet problem for the Laplace - Beltrami operator � ofthe Riemannian metric on M with the boundary values on (M) admits solution:every continuous function h� on (M) can be uniquely extended to a function hwhich is continuous on M [ (M) and harmonic on M .If in addition M as a metric space is hyperbolic in the sense of Gromov, thenthe same conclusions hold for the hyperbolic compacti�cation in the place of the endcompacti�cation.Note that the standard example of Gromov hyperbolic Riemannian manifolds isgiven by simply connected Riemannian manifolds whose negative sectional curva-tures are bounded and bounded away from zero. In this case the hyperbolic bound-ary coincides with the usual visibility boundary and convergence to the boundarywas �rst stated by Prat [Pr], whereas the solvability of the Dirichlet problem at in-�nity was proved by Anderson [An] and Sullivan [Su]. Our proof seems to be easiereven in this case (here the fact that �(M) < 0 follows from McKean's comparisontheorem for the spectrum of the Laplacian - see Chavel [Cha]), as it does not useany local di�erential geometry and rests only on two global geometric properties:hyperbolicity and Cheeger's inequality (or, equivalently, �(M) 6= 0).References[A1] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary,Ann. of Math. 125 (1987), 495{536.[A2] A. Ancona, Positive harmonic functions and hyperbolicity, Potential Theory, Surveysand Problems, (J. Kr�al, J. Luke�s, I. Netuka and J. Vesel�y, eds.) Lecture Notes in Math.1344, Springer-Verlag, Berlin - Heidelberg - New York, 1988, pp. 1{23.[An] M. T. Anderson, The Dirichlet problem at in�nity for manifolds of negative curvature,J. Di�. Geometry 18 (1983), 701{721.[A-S] M. T. Anderson and R. Schoen, Positive harmonic functions on complete manifolds ofnegative curvature, Ann. of Math. 121 (1985), 429{461.[Bi] P. Billingsley, Convergence of Probability Measures, J. Wiley and Sons, New York, 1968.[B-P] I. Benjamini and Y. Peres, Random walks on a tree and capacity in the interval, preprint(Hebrew Univ., Jerusalem) (1990).[Bu] P. Buser, A note on the isoperimetric constant, Ann. Scient. Ec. Norm. Sup. 15 (1982),213{230.[Ca] P. Cartier, Fonctions harmoniques sur un arbre, Symposia Math. 9 (1972), 203{270.[C-D-P] M. Coornaert, T. Delzant, A. Papadopoulos, Notes sur les groupes hyperboliques deGromov, Publ. IRMA Strasbourg (1989).[C-E] J.Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North Hol-land, Amsterdam, 1975.

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22 VADIM A. KAIMANOVICH AND WOLFGANG WOESS[Ve] A. M. Vershik, Multiple-valued measure preserving mapppings (polymorphisms) andMarkov operators, J. Soviet Math. 23 (1983), 2243{2266.[Wo] W. Woess, Boundaries of random walks on graphs and groups with in�nitely many ends,Israel J. Math. 68 (1989), 271{301.Department of Applied Mathematics, Leningrad Shipbuilding Institute, Lotsman-skaya 3, Leningrad 190008, Soviet UnionDipartimento di Matematica, Universit�a di Milano, Via C. Saldini 50, 20133 Mi-lano, Italia