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Pseudoprocesses, constructed by means of the solutions of higher-orderheat-type equations have been developed by several authors and many relatedfunctionals have been analyzed by means of the Feynman-Kac functional or bymeans of the Spitzer identity. We here examine even-order pseudoprocesseswrapped up on circles and derive their explicit signed density measures. Weobserve that circular even-order pseudoprocesses differ substantially frompseudoprocesses on the line because - for $t> \bar{t} > 0$, where $\bar{t}$ isa suitable $n$-dependent time value - they become real random variables. Bycomposing the circular pseudoprocesses with positively-skewed stable processeswe arrive at genuine circular processes whose distribution, in the form ofPoisson kernels, is obtained. The distribution of circular even-orderpseudoprocesses is similar to the Von Mises (or Fisher) circular normal andtherefore to the wrapped up law of Brownian motion. Time-fractional andspace-fractional equations related to processes and pseudoprocesses on the unitradius circumference are introduced and analyzed.

Even-order pseudoprocesses on a circle and related Poisson kernels