  Strong anomalous diffusion, where $\langle |x(t)|^q \rangle \sim t^{q
\nu(q)}$ with a nonlinear spectrum $\nu(q) \neq \mbox{const}$, is wide spread
and has been found in various nonlinear dynamical systems and experiments on
active transport in living cells. Using a stochastic approach we show how this
phenomena is related to infinite covariant densities, i.e., the asymptotic
states of these systems are described by non-normalizable distribution
functions. Our work shows that the concept of infinite covariant densities
plays an important role in the statistical description of open systems
exhibiting multi-fractal anomalous diffusion, as it is complementary to the
central limit theorem.

Statistical Mechanics;Chaotic Dynamics