Isometric group actions with vanishing rate of escape on $$\textrm{CAT}(0)$$ spaces

Let $\Gamma$ be a finitely generated group equipped with symmetric and nondegenerate probability measure $\mu$ finite second moment, $Y$ CAT(0) space which is either proper or of telescopic dimension. We show that if an isometric action on has vanishing rate escape respect to does not fix point in the boundary at infinity $Y$, then there exists flat subspace left invariant under $\Gamma$. In proof this result, equivariant $\mu$-harmonic map from into plays important role.