# Equivalence of harmonic measures on hyperbolic groups

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 ?