Pseudoprocesses on a circle and related Poisson kernels

摘要

Pseudoprocesses, constructed by means of the solutions of higher-order heat-type equations, have been developed by several authors and many related functionals have been analysed by applying the Feynman-Kac functional or by means of the Spitzer identity. We here examine pseudoprocesses wrapped up on circles and derive their explicit signed density measures. By composing the circular pseudoprocesses with positively skewed stable processes, we arrive at genuine circular processes whose distribution is obtained in the form of Poisson kernels. The distribution of circular even-order pseudoprocesses is similar to the Von Mises (or Fisher) circular normal law and to the wrapped up law of Brownian motion. Time-fractional and space-fractional equations related to processes and pseudoprocesses on the unit radius circumference are introduced and analysed.