On the theory of relativistic Brownian motion

TL;DR

This paper develops a theoretical framework for relativistic Brownian motion using path integrals, proposing a relativistic Wiener measure and calculating transition probabilities with consistent results across methods.

Contribution

It introduces a relativistic analogue of the Wiener measure and provides formulas for transition probabilities in relativistic Brownian motion, advancing the theoretical understanding.

Findings

Derived a formula for relativistic particle transition probabilities.

Constructed a relativistic Wiener measure as a limit of finite-difference approximations.

Obtained exact and asymptotic formulas for volumes of parts of high-dimensional cubes.

Abstract

The approach to the theory of a relativistic random process is considered by the path integral method as Brownian motion taking into account the boundedness of speed. An attempt was made to build a relativistic analogue of the Wiener measure as a weak limit of finite-difference approximations. A formula has been proposed for calculating the probability particle transition during relativistic Brownian motion. Calculations were carried out by three different methods with identical results. Along the way, exact and asymptotic formulas for the volume of some parts and sections of an N-1-dimensional unit cube were obtained. They can have independent value.