Markov Random Process on the Heisenberg Group

The paper investigates a model of the motion of a Brownian particle on \(\mathbb{C}^{2}\). A six-dimensional random process is considered, whose states are described by the coordinates of the particle and the complex analog of the area swept out by the radius vector of the point as it moves along the trajectories of a four-dimensional Brownian bridge. Using the methods of integration over the conditional Wiener measure, an expression for the transition probability density of a random process from an arbitrary state to a subsequent one is obtained. A connection is revealed between the trajectories of the process under consideration and the Heisenberg group constructed over the field of complex numbers \(H_{3}(\mathbb{C})\). The continuity in time and Hölder property with order \(\alpha<1/2\) of the oriented area function \(S(t)\) are proven, as well as the Heisenberg Markov property of the process. The heat equation describing the system’s evolution and the corresponding sub-Laplacian are derived. The solution to the equation is obtained in the form of a functional integral. Performing a Wick rotation allows for drawing an analogy between the considered process and the motion of an electron in a magnetic field.