Analytic Solutions of Equation for Random Evolution on a Complex Plane

Abstract

We discuss a generalization of Goldstein-Kac model on a complex plane and apply probabilistic approach to construct solutions of the corresponding Cauchy problem for complex-analytic initial conditions. The method is based on reconstruction of complex- analytic functions by combination of power functions, for which corresponding solutions are the moments of evolution process. As soon as in the hydrodynamic limit the equation for our model approximates a Schrödinger-type equation, the solutions constructed for pre-limit Cauchy problem may approximate solutions for corresponding Cauchy problem for a Schrödinger-type equation.