Even-order pseudoprocesses on a circle and related Poisson kernels

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.

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