Relativistic analysis of stochastic kinematics

The relativistic analysis of stochastic kinematics is developed in order to determine the transformation of the effective diffusivity tensor in inertial frames. Poisson-Kac stochastic processes are initially considered. For one-dimensional spatial models, the effective diffusion coefficient measured in a frame moving with velocity w with respect to the rest frame of the stochastic process is inversely proportional to the third power of the Lorentz factor γ (w) = (1 − w 2 /c 2 ) −1/2 . Subsequently, higher-dimensional processes are analyzed and it is shown that the diffusivity tensor in a moving frame becomes nonisotropic: The diffusivities parallel and orthogonal to the velocity of the moving frame scale differently with respect to γ (w). The analysis of discrete space-time diffusion processes permits one to obtain a general transformation theory of the tensor diffusivity, confirmed by several different simulation experiments. Several implications of the theory are also addressed and discussed.