Modern Physics Letters B, Vol. 21, No. 4 (2007) 163–174 c  World Scientific Publishing Company THE FRACTIONAL CHAPMAN KOLMOGOROV EQUATION VASILY E. TARASOV Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia tarasov@theory.sinp.msu.ru Received 4 April 2006 The Chapman–Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Frac- tional integrals can be used to describe the fractal media. Using fractional integrals, the fractional generalization of the Chapman–Kolmogorov equation is obtained. From the fractional Chapman–Kolmogorov equation, the Fokker–Planck equation is derived. Keywords : Chapman–Kolmogorov equation; fractional integral; fractal. 1. Introduction Integrals and derivatives of the fractional order goes back to Leibniz, Liouville, Rie- mann, Grunwald, and Letnikov. 1 Fractional analysis has found many applications in recent studies in mechanics and physics. The interest in fractional integrals and derivatives has been growing continually during the last few years because of nu- merous applications. In a fairly short period of time, the list of such applications has become long. It includes chaotic dynamics, 2, 3 mechanics of fractal and com- plex media, 4– 6 physical kinetics, 2, 7, 8 plasma physics, 9–11 astrophysics, 12 long-range dissipation, 13, 14 non-Hamiltonian mechanics, 15, 16 and long-range interaction. 17–19 The natural question arises: What could be the physical meaning of the frac- tional integration? This physical meaning can be following: the fractional integra- tion can be considered as an integration in some noninteger-dimensional space. If we use the well-known formulas for dimensional regularizations, 20 then we get that the fractional integration can be considered as an integration in the fractional di- mension space 15 up to the numerical factor Γ(α/2)/[2π α/2 Γ(α)]. This interpretation was suggested in Ref. 15. Fractional integrals can be considered as approximations of integrals on fractals. 21, 22 In Ref. 22, authors proved that integrals on a net of fractals can be approximated by fractional integrals. Using fractional integrals, we derive the fractional generalization of the Chapman–Kolmogorov equation. 23, 24 In this paper, the generalization of the Fokker–Planck equation for fractal media is derived from the fractional Chapman–Kolmogorov equation. In Sec. 2, a brief review of the Hausdorff measure, the Hausdorff dimension and integration on fractals is carried out to fix notation and provide a convenient 163