Markov Random Process on the Heisenberg Group

In the paper, a model of the motion of a Brownian particle on C 2 is studied. A six-dimensional random process, the states of which are described by the coordinates of the particle and a complex analogue of the area swept by the radius vector of a point as it moves along the trajectories of a four-dimensional Brownian bridge is considered. Using methods of integration over the conditional Wiener measure, it is possible to obtain an expression for the probability density of the transition of a random process from an arbitrary state to a subsequent one. A connection is revealed between the trajectories of the process under consideration and the Heisenberg group constructed over the field of complex numbers H3(C). The continuity in time and the H¨older property with order α < 1/2 of the oriented area function S(t) are proved, as well as the Heisenberg Markov property of the process. The heat equation describing the evolution of the system and the corresponding sub-Laplacian are found. The solution to the equation is obtained in the form of a functional integral. The performance of a Wick rotation allows to draw an analogy between the process under consideration and the movement of an electron in a magnetic field.