Fractional Generalization of Kac Integral

Kac integral [1, 2, 3] appears as a path-wise presentation of Brownian motion and shortly becomes, with Feynman approach [4], a powerful tool to study different processes described by the wave-type or diffusion-type equations. In the basic papers [1, 4], the paths distribution was based on averaging over the Wiener measure. It is worthwhile to mention the Kac comment that the Wiener measure can be replaced by the Lévy distribution that has infinite second and higher moments. There exists a fairly rich literature related to functional integrals with generalization of the Wiener measure (see for example [5, 6]). Recently the Lévy measure was applied to derive a fractional generalization of the Schrödinger equation [7, 8] using the Feynman-type approach and expressing the Lévy measure through the Fox function [9] In this paper, we derive the fractional generalization of the diffusion equation (FDE) from the path integral over the Lévy measure using the integral equation approach of Kac.