NMR survey of reflected Brownian motion

Restricted diffusion is a common feature of many physicochemical, biological, and industrial processes. Nuclear magnetic resonance techniques are often used to survey the atomic or molecular motion in confining media by applying inhomogeneous magnetic fields to encode the trajectories of spin-bearing particles. The diversity and complexity of diffusive NMR phenomena, observed in experiments, result from the specific properties of reflected Brownian motion. Here the focus is on the mathematical aspects of this stochastic process, their physical interpretations, and their practical applications. The main achievements in this field, from Hahn's discovery of spin echoes to present-day research, are presented in a unified mathematical language. A long-standing problem of restricted diffusion under arbitrary magnetic field is reformulated in terms of multiple correlation functions of the reflected Brownian motion. Many classical results are retrieved, extended, and critically discussed.