Relativistic Brownian Motion

We solve the problem of formulating Brownian motion in a relativistically covariant framework in 1+1 and 3+1 dimensions. We obtain covariant Fokker-Planck equations with (for the isotropic case) a differential operator of invariant d'Alembert form. Treating the spacelike and timelike fluctuations separately, we show that it is essential to take into account the analytic continuation of "unphysical" fluctuations in order to achieve these results.