Dimension of harmonic measures in hyperbolic spaces

Abstract

We show exact dimensionality of harmonic measures associated with random walks on groups acting on a hyperbolic space under finite first moment condition, and establish the dimension formula by the entropy over the drift. We also treat the case when a group acts on a non-proper hyperbolic space acylindrically. Applications of this formula include continuity of the Hausdorff dimension with respect to driving measures and Brownian motions on regular coverings of a finite volume Riemannian manifold.

Harmonic measures on Bowditch boundaries of groups hyperbolic relative to virtually nilpotent subgroups

“…Next, we show that the harmonic measure for (X, P, m) is exact dimensional, that is, the ratio of the logarithm of the harmonic measure of a ball and the logarithm of the radius of a ball (in suitable visual metrics) has a limit as the radius approaches zero (a detailed discussion of the literature on this topic in related contexts appears in Tanaka [48]). The formula for the dimension can be given in terms of drift and asymptotic entropy.…”

“…The main issue in our case is to extend the shadow lemma to the shadow of balls centered at non-orbit points. Another approach from Kaimanovich [32], Le Prince [38], Tanaka [48], applicable rather generally for groups acting with exponential growth on proper hyperbolic spaces and jumps given by measures with finite first moment (see also Dussaule-Yang [16] for harmonic measures of random walks on Cayley graphs of relatively hyperbolic groups), utilizes properties of the asymptotic entropy, drift (Kaimanovich [33]; Maher-Tiozzo [40] in the non-proper case) in the hyperbolic setting and the ergodicity of the action of Γ with respect to the harmonic measure in the boundary (for the lower bound on the dimension, [48]). The dimension of µ, the D Γ -dimensional Patterson-Sullivan density of Γ for its action on X, where D Γ is the critical exponent of Γ acting on X (µ being unique up to constant multiples as it is non-atomic, doubling and ergodic for the Γ-action, see Remark 4.3 and Remark 4.12 below) is computed by an approach of Stratmann-Velani [46], using a shadow lemma to study cusp excursions into horoballs.…”

“…The question of ascertaining the dimension of the harmonic measure in related settings has been treated in Ledrappier [35,37], Kaimanovich [32], Le Prince [38], [5], Tanaka [48]. It has been treated for random walks on relatively hyperbolic groups in [16], for harmonic measures in both Floyd and Bowditch boundaries.…”

On the speed of distance-stationary sequences

“…Letting p to be equal to 1, and combining the previous estimate with the relationship between entropy, dimension and speed (see for example Tanaka, 2019, Hochman and Solomyak, 2017, and Ledrappier, 1984 the following dimension drop result for the escape measure is obtained:…”

Entropy and drift for Gibbs measures on geometrically finite manifolds

“…These quantities are related via the inequality (1) h µ ď ℓ µ v due to Guivarc'h [23] (see also Vershik [42], who calls it the fundamental inequality). Let us remark that h l is the Hausdorff dimension of the harmonic measure [41]. On the other hand, to a random walk on Γ we can associate a Green metric d G on Γ, defined in [2], with d G pg, hq defined as the negative logarithm of the probability that a random path starting at g ever hits h. A measure µ is generating Γ if the semigroup generated by the support of µ equals Γ.…”