Relativistic diffusion processes and random walk models
Abstract
The nonrelativistic standard model for a continuous, one-parameter diffusion process is the Wiener process. As well-known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate at velocities larger than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular diffusion fronts on the light cone, which are unlikely to exist for massive particles. Attempting to avoid such singularities, we discuss non-Markovian relativistic generalizations of the Wiener process by focussing directly on the desired properties of the transition PDFs. This approach yields two types of continuous diffusion models for massive particles which could be used as a viable alternative to the solutions of the telegraph equation.Keywords
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