Even-order pseudoprocesses on a circle and related Poisson kernels

TL;DR

This paper investigates even-order pseudoprocesses on a circle, deriving explicit signed densities, and explores their connections to Poisson kernels, stable processes, and fractional equations, revealing their distinct behavior from line pseudoprocesses.

Contribution

It provides explicit density measures for circular even-order pseudoprocesses and links them to Poisson kernels and fractional equations, highlighting their unique properties.

Findings

Circular pseudoprocesses become real random variables after a certain time.

Composition with stable processes yields genuine circular processes with Poisson kernel distributions.

Distribution resembles the Von Mises circular normal, akin to wrapped Brownian motion.

Abstract

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 analyzed by means of the Feynman-Kac functional or by means of the Spitzer identity. We here examine even-order pseudoprocesses wrapped up on circles and derive their explicit signed density measures. We observe that circular even-order pseudoprocesses differ substantially from pseudoprocesses on the line because - for $t> \bar{t} > 0$, where $\bar{t}$ is a suitable $n$-dependent time value - they become real random variables. By composing the circular pseudoprocesses with positively-skewed stable processes we arrive at genuine circular processes whose distribution, in the form of Poisson kernels, is obtained. The distribution of circular even-order pseudoprocesses is similar to the Von Mises (or Fisher) circular normal and therefore 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 analyzed.