Functional Feynman-Kac Equations for Limit Lognormal Multifractals

A novel technique of functional Feynman-Kac equations is developed for the probability distribution of the limit lognormal multifractal process introduced by Mandelbrot [in Statistical Models and Turbulence, M. Rosenblatt and C. Van Atta, eds., Springer, New York (1972)] and constructed explicitly by Bacry, Delour, and Muzy [Phys. Rev. E 64:026103 (2001)]. The distribution of the process is known to be determined by the complicated stochastic dependence structure of its increments (SDSI). It is shown that the SDSI has two separate layers of complexity that can be captured in a precise way by a pair of functional Feynman-Kac equations for the Laplace transform. Exact solutions are obtained as power series expansions in the intermittency parameter using a novel intermittency differentiation rule. The expansion of the moments gives a new representation of the Selberg integral. The author wishes to express gratitude to the Mathematics Department of Lehigh University for generous support during his stay at Lehigh University, where this article was written.