hadamard (or complete $cat(0)$) spaces are complete, non-positive curvature, metric spaces. here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. our results extend the standard non-linear ergodic theorems for non-expansive maps on real hilbert spaces, to non-expansive maps on had...

We discuss when law-invariant convex functionals “collapse to the mean”. More precisely, we show that, in a large class of spaces random variables and under mild semicontinuity assumptions, expectation functional is, up an affine transformation, only that is linear along direction nonconstant variable with nonzero expectation. This extends results obtained literature bounded setting additional ...

We show that, for a (not necessarily continuous) weakly contractive mean-type mapping $\mathbf{M} \colon I^p\to I^p$ (where $I$ is an interval and $p \in \mathbb{N}$), the functional equation $K \circ \mathbf{M}=K$ has at most one solution in family of continuous means I^p \to I$. Some general approach to latter also given.