Absolute continuity of harmonic measure for a random walk and its reflection

Question

Let $G$ be a hyperbolic group, and $\mu$ a (nonsymmetric) probability measure on $G$ whose support generates $G$ as a semigroup. Let $\nu$ be the associated harmonic ($\mu$ stationary) on $\partial G$. Let $\hat{\mu}$ be the reflected random walk given by $\hat{\nu}(g)=\mu(g^{-1})$. Let $\hat{\nu}$ be the $\hat{\mu}$ harmonic measure on $\partial G$. My question is: are $\nu$ and $\hat{\nu}$ absolutely continuous?

Some intuition: Ryokichi Tanaka in https://arxiv.org/pdf/1411.2312.pdf showed that under the assumotion that $\nu$ has finite support, $\nu$ is absolutely continuous to the Patterson-Sullivan measure if and only if $\hat{\nu}$ is. The argument seems to be that the ratio of the drift and asymptotic entropy is the same (equal to growth rate of the group) for both measures, and this should imply that their Green metrics are comparable and so the harmonic measures are absolutely continuous. However, Tanaka's result heavily relies on the Ancona inequalities which are only available when the measure has superexponential moment. I am wondering if there is some softer argument to approach this problem which holds under weaker moment conditions, or in a situation where for instance $G$ is not a hyoerbolic group but a subgroup of isometries of a Gromov hyperbolic space.

Answer 1

In general there is no reason for the coincidence of the measure classes of the harmonic measures of the original and of the reflected random walks. It they do coincide, then this indicates that the random walk in question must be in a certain respect "special'', and it is very easy to construct counterexamples, already for free groups. In Tanaka's result which you quote the argument is essentially based on the unqueness of a maximal entropy Gibbs measure for subshifts of finite type.

EDIT. The key property that one should always keep in mind is the following fact relating the entropy, rate of escape, and the Hausdorff dimension of the harmonic measure. Let $\mu$ be a "nice" measure on a hyperbolic group $G$, let $d$ be a translation invariant hyperbolic metric on $G$ quasi-isometric to the word metric, and let $\rho$ be the associated metric on the boundary $\partial G$. Then $$ \operatorname{HD}_\rho \nu = \frac{h}{\ell} \;, $$ where $\nu$ is the harmonic measure on the boundary, $\operatorname{HD}_\rho\nu$ is its Hausdorff dimension with respect to $\rho$, $h$ is the asymptotic entropy of the random walk, and $\ell$ is the rate of escape with respect to $d$. Now, the dimension of any measure does not exceed the dimension of its support, so that $$ \operatorname{HD}_\rho\nu \le \operatorname{HD}_\rho \partial G \;, $$ and $\operatorname{HD}_\rho \partial G$ coincides with the exponential growth rate $v$ of the group $G$ with respect to the distance $d$.

The other fact is the "uniqueness of the maximal entropy measure". In our situation it means that there is a certain collection of measure classes (in the classical negatively curved setup these are the measure classes of Gibbs measures) which contains the classes of the Hausdorff measures of "nice" boundary metrics $\rho$ and the classes of the harmonic measures $\nu$ of "nice" random walks, and such that for any such metric $\rho$ there is only one measure class from this collection (namely, the class of the Hausdorff measure of $\rho$) whose Hausdorff dimension coincides with $\operatorname{HD}_\rho \partial G$.

Thus, if $h/\ell=v$, then the harmonic measure $\nu$ must be equivalent to the Hausdorff measure of the metric $\rho$ (i.e., to the Patterson measure of the distance $d$ on the group). It is this latter property that is used several times in the BHM paper (first, for a general distance on $G$, and then for the Green distance determined by another random walk). However, if $h/\ell<v$, there is no uniqueness, and one can have lots of different measure classes of the same dimension.

For a concrete example, take the free group with 2 generators and a non-symmetric measure supported by the generating set. Then there is an explicit formula for the resulting harmonic measure (going back to at least Dynkin--Malyutov 1961 paper), from which one can easily see that the harmonic measures corresponding to different step distributions are all pairwise singular. For a very concrete example suppose that $\mu$ is almost entirely concentrated just on a single generator $a$; then the associated harmonic measure will be concentrated on infinite words in which the frequency of $a$ is close to 1. Obviously, the reflected measure $\check\mu$ has the same property with respect to $a^{-1}$, so that the harmonic measures of $\mu$ and $\check\mu$ are singular.