  We study dynamical properties of random Schr\"odinger operators
$H^{(\omega)}$ defined on the Hilbert space $\ell^2(\bbZ^d)$ or $L^2(\bbR^d)$.
Building on results from existing multi-scale analyses, we give sufficient
conditions on $H^{(\omega)}$ to obtain the vanishing of the diffusion exponent
$$
  \sigma_{\rm diff}^+ := \limsup_{T\rightarrow\infty } \frac{\log
  \bbE \left(\la\la\vert X
  \vert^2\ra\ra_{T,f_I(H^{(\omega)})\psi}\right) }{\log T}=0. $$ Here $\bbE$ is
the expectation over randomness, $f_{I}$ is any smooth characteristic function
of a bounded energy-interval $I$ and $\psi$ is a state vector in the domain of
$H^{(\omega)}$ with compact spatial support. The quantity $\la\la |X|^2
\ra\ra_{T,\varphi}$ denotes the Cesaro mean up to time $T$ of the second moment
of position $\la |X|^2\ra_{t,\varphi}$ at times $0\le t\le T$ of an initial
state vector $\varphi$. If the Hilbert space is $\ell^2(\bbZ^d)$, the method of
proof can be strengthened to yield dynamical localization. Under weaker
assumptions, we also prove a theorem on the absence of diffusion. The results
are applied to a randomly perturbed periodic Schr\"odinger operator on
$L^2(\bbR^d)$, to a simple Anderson-type model on the lattice and to a model
with a correlated random potential in continuous space.
