  We study the thermal Markovian diffusion of tracer particles in a 2D medium
with spatially-varying diffusivity $D(r)$, mimicking recently measured,
heterogeneous maps of the apparent diffusion coefficient in biological cells.
For this heterogeneous diffusion process (HDP) we analyse the mean squared
displacement (MSD) of the tracer particles, the time averaged MSD, the spatial
probability density function, and the first passage time dynamics from the cell
boundary to the nucleus. Moreover we examine the non-ergodic properties of this
process which are important for the correct physical interpretation of time
averages of observables obtained from single particle tracking experiments.
From extensive computer simulations of the 2D stochastic Langevin equation we
present an in-depth study of this HDP. In particular, we find that the MSDs
along the radial and azimuthal directions in a circular domain obey anomalous
and Brownian scaling, respectively. We demonstrate that the time averaged MSD
stays linear as a function of the lag time and the system thus reveals a weak
ergodicity breaking. Our results will enable one to rationalise the diffusive
motion of larger tracer particles such as viruses or submicron beads in
biological cells.
