  Brownian motion is widely used as a paradigmatic model of diffusion in
equilibrium media throughout the physical, chemical, and biological sciences.
However, many real world systems, particularly biological ones, are
intrinsically out-of-equilibrium due to the energy-dissipating active processes
underlying their mechanical and dynamical features. The diffusion process
followed by a passive tracer in prototypical active media such as suspensions
of active colloids or swimming microorganisms indeed differs significantly from
Brownian motion, manifest in a greatly enhanced diffusion coefficient,
non-Gaussian tails of the displacement statistics, and crossover phenomena from
non-Gaussian to Gaussian scaling. While such characteristic features have been
extensively observed in experiments, there is so far no comprehensive theory
explaining how they emerge from the microscopic active dynamics. Here we
present a theoretical framework of the enhanced tracer diffusion in an active
medium from its microscopic dynamics by coarse-graining the hydrodynamic
interactions between the tracer and the active particles as a stochastic
process. The tracer is shown to follow a non-Markovian coloured Poisson process
that accounts quantitatively for all empirical observations. The theory
predicts in particular a long-lived L\'evy flight regime of the tracer motion
with a non-monotonic crossover between two different power-law exponents. The
duration of this regime can be tuned by the swimmer density, thus suggesting
that the optimal foraging strategy of swimming microorganisms might crucially
depend on the density in order to exploit the L\'evy flights of nutrients. Our
framework provides the first validation of the celebrated L\'evy flight model
from a physical microscopic dynamics.
