Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, the random walk $X_n$ driven by $\mu$ can visit the whole group $\Gamma$.

Fact : the random walk $X_n$ almost surely converges to a point in the Gromov boundary $\partial X$ of $X$. Let $\nu$ be the exit measure on $\partial X$. Then, $(\partial X,\nu)$ is a model for the so-called Poisson boundary. The measure $\nu$ is called the harmonic measure (with respect to $\mu$).

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

Question : Are $\nu$ and $\check{\nu}$ equivalent ?