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The Feynman-Kac equations are a type of partial differential equationsdescribing the distribution of functionals of diffusive motion. The probabilitydensity function (PDF) of Brownian functionals satisfies the Feynman-Kacformula, being a Schr\"{o}dinger equation in imaginary time. The functionals ofno-Brownian motion, or anomalous diffusion, follow the fractional Feynman-Kacequation [J. Stat. Phys. 141, 1071-1092, 2010], where the fractionalsubstantial derivative is involved. Based on recently developed discretizedschemes for fractional substantial derivatives [arXiv:1310.3086], this paperfocuses on providing algorithms for numerically solving the forward andbackward fractional Feynman-Kac equations; since the fractional substantialderivative is non-local time-space coupled operator, new challenges areintroduced comparing with the general fractional derivative. Two ways (finitedifference and finite element) of discretizing the space derivative areconsidered. For the backward fractional Feynman-Kac equation, the numericalstability and convergence of the algorithms with first order accuracy aretheoretically discussed; and the optimal estimates are obtained. For all theprovided schemes, including the first order and high order ones, of bothforward and backward Feynman-Kac equations, extensive numerical experiments areperformed to show their effectiveness.

Numerical algorithms for the forward and backward fractional Feynman-Kac equations