  Diffusivity is a key quantity in describing velocity fluctuations in granular
materials. These fluctuations are the basis of many thermodynamic and
hydrodynamic models which aim to provide a statistical description of granular
systems. We present experimental results on diffusivity in dense, granular
shear in a 2D Couette geometry. We find that self-diffusivities are
proportional to the local shear rate with diffusivities along the mean flow
approximately twice as large as those in the perpendicular direction. The
magnitude of the diffusivity is D \approx \dot\gamma a^2 where a is the
particle radius. However, the gradient in shear rate, coupling to the mean
flow, and drag at the moving boundary lead to particle displacements that can
appear sub- or super-diffusive. In particular, diffusion appears superdiffusive
along the mean flow direction due to Taylor dispersion effects and subdiffusive
along the perpendicular direction due to the gradient in shear rate. The
anisotropic force network leads to an additional anisotropy in the diffusivity
that is a property of dense systems with no obvious analog in rapid flows.
Specifically, the diffusivity is supressed along the direction of the strong
force network. A simple random walk simulation reproduces the key features of
the data, such as the apparent superdiffusive and subdiffusive behavior arising
from the mean flow, confirming the underlying diffusive motion. The additional
anisotropy is not observed in the simulation since the strong force network is
not included. Examples of correlated motion, such as transient vortices, and
Levy flights are also observed. Although correlated motion creates velocity
fields qualitatively different from Brownian motion and can introduce
non-diffusive effects, on average the system appears simply diffusive.
