  We study a nonlinear semigroup associated to a nonexpansive mapping on a
Hadamard space and establish its weak convergence to a fixed point. A
discrete-time counterpart of such a semigroup, the proximal point algorithm,
turns out to have the same asymptotic behavior. This complements several
results in the literature -- both classical and more recent ones. As an
application, we obtain a new approach to heat flows in singular spaces for
discrete, as well as continuous times.
