Fractional generalization of Kac integral
\textit{M. Kac}'s integral [Bull. Am. Math. Soc. 54, 64 (1948); Trans. Am. Math. Soc. 65, 1--13 (1949; Zbl 0032.03501)] appeared as a path-wise presentation of Brownian motion, and with \textit{R.P. Feynman}'s approach [Rev. Mod. Phys. 20, 367--387(1948)], became a powerful tool to study different processes described by the wave-type or diffusion-type equations. Recently the Lévy measure was applied to derive a fractional generalization of the Schrödinger equation, using the Feynman-type approach and expressing the Lévy measure through the Fox \(H\)-function. In this paper the authors derive the fractional generalization of the diffusion equation from the path integral over the Lévy measure using the integral equation approach of Kac. A function introduced in the analysis is expressed in terms of Fox \(H\)-function.
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