Some aspects of the functional integral approach to relativistic Brownian motion

The report investigates the possibility of describing the transition probability of relativistic Brownian motion by means of a functional integral in the configuration space. It is shown that the sequence of finite-time integrals corresponding to the functional integral with the action of a free relativistic particle converges to the delta function in the space of generalized functions. This result is extended to functional integrals of general position with bounded wandering rate. Thus, the description of the relativistic Brownian motion by a functional integral in the configuration space is untenable, but such a description is possible in the phase space. A regularization method for the functional integral with the action of relativistic particle is considered and estimates for the limit are obtained. The report is based on joint work with I.V. Volivich and E.A. Kurianovich.