A Relation between Dimension of the Harmonic Measure, Entropy and Drift for a Random Walk on a Hyperbolic Space

We establish in this paper an exact formula which links the dimension of the harmonic measure, the asymptotic entropy and the rate of escape for a random walk on a discrete subgroup of the isometry group of a Gromov hyperbolic space. This completes a result of [11], where only an upper bound for the dimension was proved.


Introduction
Let Γ be a discrete subgroup of the isometry group of a Gromov hyperbolic space X and µ a probability measure on Γ.The associated (right) random walk is the processus (x n ) n≥0 defined by: x 0 = e and, for n ≥ 1, x n = h 1 • • • h n , where (h i ) i>0 is a sequence of independent random variables which take their values in Γ, with distribution µ.Under some additional hypotheses, a point o in X being fixed, the trajectory (x n o) n≥0 converges almost surely to a point x ∞ in the geometric boundary ∂X of X.The distribution ν of x ∞ is called the harmonic measure associated with the random walk.
The boundary ∂X is equipped with a family of metrics d a , where a is a parameter.The aim of this paper is to established the following formula, which links the dimension of ν with respect to d a , the asymptotic entropy h(µ) and the rate of escape l(µ) of the random walk (see Theorem 3.1 for a precise statement): This result completes the result obtained in [11], where only the bound from above in (0.1) was proved.The main goal of the above mentioned paper was to show that the harmonic measure could be singular with respect to the Hausdorff measure; we refer to it for additional comments on the question of the dimension of the harmonic measure.
The first result on this subject was obtained in [10] for a random walk on SL(2, R) (which acts by isometries on the hyperbolic disc); a similar result was established in the case where X is a tree in [8].In our context of a hyperbolic space, and under the assumptions that µ is symmetric and finitely supported, the following result on the pointwise dimension is established in [2]: The dimension involved in our main theorem (3.1) is the box dimension and in this sense our result is weaker than (0.2). Indeed when log νB a (ξ, r)/log r converges a.s. to a constant, all reasonable notions of dimension coincide (see [13] or [12]).The interest of our result however lies in the fact that our hypotheses are the weakest possible -i.e. the weakest under which (0.1) makes sense: we just assume that µ admits a finite first moment.
The paper is organized as follows.In Section 1 we recall briefly the needed notions on random walks, hyperbolic spaces and dimensions of measures.We prove also in this first section a covering result on ∂X which is necessary in order to compare the dimensions.In Section 2 we establish the lower bound for the dimension; which allows us to state our main result in Section 3.
1 Preliminaries and notations

Hyperbolic space
The notion of Gromov hyperbolic space goes back to [7].We refer to [4] and [6] for the definitions and properties that we state in this paragraph.

Definition
Let (X, d) be a Gromov hyperbolic space.We denote by B(x, R) the open ball with center x and radius R in (X, d).Recall that the Gromov product (w.r.t. a point w ∈ X) on X 2 is defined by: The fact that (X, d) is δ-hyperbolic, for δ ≥ 0, is characterized by the fact that for every w ∈ X, for every (x, y, z) We assume moreover that (X, d) is proper and geodesic; we fix a base point o in X and denote by (.|.) the Gromov product with respect to o.

Boundary
The space X is compactified by its geometric boundary ∂X.The action of the isometry group on X extends to a continuous action on the boundary.We also denote by (.|.) the extension of the Gromov product to ∂X; on X ∪ ∂X, (1.2) becomes: Two distinct points of X ∪ ∂X can be joined by a geodesic.We shall also use the fact that for x, y ∈ X ∪ ∂X, if (x n ) n≥0 converges to x and (y n ) n≥0 converges to y, then On this boundary a family of metrics d a , 1 < a < a 0 , is defined, which have the following property: there exists a constant λ > 1 such that ∀x, y ∈ ∂X We denote by B a (ξ, r) the open balls in (∂X, d a ); and, for r > 0, log a r = log r log a .

Discrete subgroups
Let Γ be a group of isometries acting properly discontinuously on X.In particular, each orbit of Γ in X is discrete.Denote by L Γ the limit set of Γ.The group Γ is said to be elementary if Another property we shall need is quasi-convex-cocompactness of Γ, which admits the following characterization ( [3]): there exists C > 0 such that every geodesic radius σ joining o to a point in L Γ stay at a distance lower than C from Γo.Note that if v(Γ) is finite and in addition Γ is quasi-convex-cocompact, then the Hausdorff dimension (with respect to the metric d a ) of L Γ is equal to v(Γ)/ log a (see [3]).
As important examples of quasi-convex-cocompact discrete groups, let us quote finitely generated hyperbolic groups acting on their Cayley graph (w.r.t. a finite set of generators) equipped with the word-length metric, or discrete convex cocompact groups of isometries acting on Riemaniann manifold with pinched sectional curvature.

Definition
Given a countable group Γ and a probability measure µ on Γ, the random walk associated to µ is defined as the homogeneous Markov processus on Γ with initial state e and transition probability p(g, h) = µ(g −1 h).Given a sequence (h n ) n≥1 of independent random variables with law µ, one defines a realization x x x = (x n ) n≥0 of this processus by x 0 = e and for n > 0, is called the trajectory of the random walk.We denote by P the law of this trajectory in Γ N .The distribution of the position x n of the trajectory at time n is the convolution product µ n = µ * • • • * µ, where µ appears n times.

Asymptotic entropy
Recall that the entropy of a probability measure m with countable support is defined by: The sequence (H(µ n )) n is subadditive so, assuming that the entropy of µ is finite, one can define the asymptotic entropy ( [1]) of the random walk by Noting that and using Kingman subadditive ergodic theorem, one gets ( [5]) a P-almost sure convergence:

Rate of escape
Consider now a group Γ acting by isometries on the metric space (X, d).Assume that µ admits a finite first moment, that is to say The rate of escape (also called drift) l(µ) of the random walk is defined, using once again a subadditivity property, by Note that under the assumption that v(Γ) < +∞, if µ admits a finite first moment, then it has also a finite entropy (see [9]), and so the asymptotic entropy of the random walk is well defined.
From now on, and unless otherwise specified, Γ is a non-elementary group acting by isometries on a proper geodesic δ-hyperbolic space (X, d).We fix a metric d a on the boundary.We assume that Γ acts properly discontinuously, is quasi-convex-cocompact, and that its exponential growth rate is finite.Moreover µ is a fixed probability measure on Γ, such that its support generates Γ as a semigroup and which admits a finite first moment.We consider the random walk (x n ) n≥0 associated with µ ; and we adopt the above notations.

Asymptotic behaviour of the random walk
Under the above hypotheses, the behaviour of the paths of the random walk is described by the following: We denote by bnd the map (defined on a set of P-measure 1) from Γ N to ∂X which associates x ∞ to x x x = (x n ) n≥0 , and by ν the distribution of x ∞ , which we call the harmonic measure.In particular we have ν = bnd(P).
The support of ν is clearly included in the limit set L Γ .It can be using the minimality of L Γ and the fact that the measure ν is µ-stationary, that the support of ν is in fact equal to L Γ (see e.g.[11]).

Dimension of measures
The Hausdorff dimension of a probability measure m on ∂X is defined by Note that this dimension is an invariant of the type of the measure m.Now let us define other notions of dimension of a measure we shall use and explore the relations between them (see [12] for an extensive presentation of this subject).
The first is the so-called box dimension.We denote by N (A, r) the minimal number of balls of radius r needed to cover a set A ⊂ ∂X.The lower and upper box dimensions of A are defined respectively as When the upper and lower box dimensions coincide, we denote it by dim B m.
We focus now on the pointwise dimension.The lower and upper pointwise dimensions are defined for every ξ in X respectively by: dim P m(ξ) = lim inf r→0 log mB a (ξ, r) log r and dim P m(ξ) = lim sup r→0 log mB a (ξ, r) log r , In order to compare the box and pointwise dimensions, we shall need a covering property on the limit set.Recall that a cover of a set A is said to have finite multiplicity M if each point of A lies in at most M elements of the cover.For a point ξ ∈ ∂X, we denote by σ ξ a (unit speed) geodesic ray joining o to ξ.For the rest of this paragraph, we omit to precise that the properties are satisfied for r small enough.The following lemma establishes a link between shadows and balls on ∂X.
Proposition 1.3.There exists a constant M > 0 such that every subset A of L Γ admits, for every r > 0, a covering by balls centered in A, of radius r, with multiplicity M .
Proof.Let A be a subset of L Γ and r > 0. We set where the constant C comes from the caracterization of quasi-convex-cocompactness (see par. 1.1).Since Γ is quasi-convex-cocompact, for each ξ ∈ A, there exists g r,ξ ∈ Γ such that Denote by {g i , i ∈ I} the set of g r,ξ , ξ ∈ A, and for each i ∈ I choose ξ i ∈ A such that g r,ξi = g i .Now, for a ξ ∈ A, take i ∈ I such that g r,ξ = g i .We have So by the use Lemma 1.2, and ξ ∈ B a (ξ i , r).Therefore {B a (ξ i , r), i ∈ I}, is a covering of A. Now let us prove that this covering has finite multiplicity (not depending on r).For ξ ∈ ∂X, let be i ∈ I such that ξ ∈ B a (ξ i , r).We have, by the second part of Lemma 1.2, d (σ ξ (T ), σ ξi (T )) < 2 log a λ + 4| log a λ + C + 4δ| + 8δ , and so where D is a constant (i.e.does not depend on r but only on C, δ and a).But the number of elements in Γ that are at a distance lower than D from σ ξ (T ) is bounded by a constant M , so that the covering just constructed has finite multiplicity M .
This covering property of the boundary has the following consequence (see [12]): Proposition 1.4.Assume that for a probability measure m on L Γ , there exists d such that dim P m(ξ) ≤ d for m-almost every ξ.Then dim B m ≤ d.
We have also (see [12]) dim H m ≤ dim B m.
2 Lower bound for the dimension Proposition 2.1.The harmonic measure ν associated to the random walk satisfies .
In this section, in order to simplify the notations, we write d for the dimension dim B ν and l and h for respectively l(µ) and h(µ).We first need the following lemma, which is a direct consequence of the definition of this dimension.
Lemma 2.2.For every η > 0, there exists a sequence (r k ) k>0 which converges to zero and such that for every k > 0, there exists a set E η,k ⊂ ∂X which satisfies: Now let us prove Proposition 2.1.Fix ǫ > 0, η > 0 and set where σ bnd(x x x) denotes a geodesic joining o to x ∞ = bnd(x x x).In view of the behaviour of the random walk (Theorem 1.1 and (1.7)), there exists N ǫ,η , such that for every n ≥ N ǫ,η , P(Ω n ǫ ) ≥ 1 − η.Then we take (r k ) k>0 as in Lemma 2.2, set n k = [(− log a r k )/l] (integer part) and, with the notations of this lemma, We immediately get, for k such that n k ≥ N ǫ,η , P(Π k ǫ,η ) ≥ 1 − 2η, and Let us fix ξ ∈ E η,k and assume that x x x ∈ Ω n k ǫ ∩ bnd −1 (B a (ξ, r k )).We have d(bnd(x x x), ξ) < r k , so that, by Lemma 1.2, Finaly we note that lim k→∞ (log a r k )/l n k = 1; so, doing k → ∞, and since ǫ and η are arbitrary, this achieves the proof.

Conclusion
It was shown in [11] that in our context, the harmonic measure satisfies: Remark 3.2.Under our hypotheses, it is known that h(µ) > 0 (see [9]); so as a consequence of Theorem 3.1 we have dim B ν > 0.
r→0 log N (A, r) log 1/r and dim B A = lim sup r→0 log N (A, r) log 1/r , and the lower upper box dimensions of a probability measure m as dim B m = lim η→0 inf{dim B A | m(A) ≥ 1 − η} dim B m = lim η→0 inf{dim B A | m(A) ≥ 1 − η} .

3 . 1 .
ν − a.s. .So, taking into account the result of Section 2 and Proposition 1.4, we get the following Theorem Under our hypotheses, the harmonic measure satisfies: