Consider a hyperbolic group Gromov $ Gamma $ and let $ mu $ be a measure of probability finely supported by $ Gamma $. Suppose the support of $ mu $ generate $ Gamma $ as a semi-group, in other words, the random walk $ X_n $ conducted by $ mu $ you can visit the whole group $ Gamma $.

Fact: the random walk $ X_n $ almost certainly converges to a point on the limit of Gromov $ partial X $ from $ X $. Leave $ nu $ be the output measure in $ partial X $. So, $ ( partial X, nu) $ It is a model for the so-called Poisson limit. Measure $ nu $ it is called the harmonic measure (with respect to $ mu $)

Now consider the inverse measure $ check { mu} $ defined by $ check { mu} (g) = mu (g -1) $ and let $ check { nu} $ be the corresponding harmonic measure.

**Question** : They are $ nu $ Y $ check { nu} $ equivalent?

It seems that the answer is positive. In the document Harmonic measures versus quasi-formal measures for hyperbolic groups, Blachère, Haïssinsky and Mathieu demonstrate that $ h = lv $ in a hyperbolic group if and only if the harmonic measure $ nu $ and the Patterson-Sullivan measure $ rho $ are equivalent here, $ h $, is asymptotic entropy, $ l $ is asymptotic drift and $ v $ It is the growth of the volume. Very roughly, the test is like that.

First, both measures are ergodic, so they are equivalent or mutually unique. The Hausdorff dimension of $ nu $ is $ h / l $ and the Hausdorff dimension of $ rho $ is $ v $. Using Lebesgue's differentiation theorem, they show that they are equivalent if and only if they have the same Hausdorff dimension. This part is very technical, the basic ingredients being the shadow slogans for the Patterson-Sullivan measurement and for the harmonic measurement.

From shadow slogans to harmonic measurements $ nu $ Y $ check { nu} $ wait and from $ nu $ Y $ check { nu} $ they have the same Hausdorff dimension, I think you can adapt your test to show that $ nu $ Y $ check { nu} $ they are equivalent

However, I would like to try a similar result in a wider environment and this strategy seems too complicated and too technical. Then I wonder if there is a much simpler test.

Keep in mind that one really needs to use hyperbolicity somewhere. For example, consider a non-centered probability measure $ mu $ in $ mathbb {Z} ^ d $, that is to say

$$ p = sum_ {x in mathbb {Z} ^ d} x mu (x) neq 0. $$

So, $ p $ It is called the drift of the random walk. According to Ney and Spitzer's theorem, Martin's limit of the random walk is a sphere: a sequence of points $ x_n $ converges to a point at the limit if and only if $ x_n $ goes to infinity and $ x_n $ converges in direction, that is $ x_n / | x_n | $ converge The harmonic measure rests in the direction given by $ p $. In particular, the harmonic measure associated with $ check { nu} $ it's compatible with $ -p $ and so $ nu $ Y $ check { nu} $ they are unique