Diffusion at solid-liquid interfaces is crucial in many technological and biophysical processes. Although its behavior seems deceivingly simple, recent studies showing passive superdiffusive transport suggest diffusion on surfaces may hide rich complexities. In particular, bulk-mediated diffusion occurs when molecules are transiently released from the surface to perform three-dimensional excursions into the liquid bulk. This phenomenon bears the dichotomy where a molecule always return to the surface but the mean jump length is infinite. Such behavior is associated with a breakdown of the central limit theorem and weak ergodicity breaking. Here, we use single-particle tracking to study the statistics of bulk-mediated diffusion on a supported lipid bilayer. We find that the time-averaged mean square displacement (MSD) of individual trajectories, the archetypal measure in diffusion processes, does not converge to the ensemble MSD but it remains a random variable, even in the long observation-time limit. The distribution of time averages is shown to agree with a L{e}vy flight model. Our results also unravel intriguing anomalies in the statistics of displacements. The time averaged MSD is shown to depend on experimental time and investigations of fractional moments show a scaling $langle |r(t)|^qrangle sim t^{q u(q)}$ with non-linear exponents, i.e. $ u(q) eqtextrm{const}$. This type of behavior is termed strong anomalous diffusion and is rare among experimental observations.
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