  The fractional Feynman-Kac equations describe the distribution of functionals
of non-Brownian motion, or anomalous diffusion, including two types called the
forward and backward fractional Feynman-Kac equations, where the fractional
substantial derivative is involved. This paper focuses on the more widely used
backward version. Based on the discretized schemes for fractional substantial
derivatives proposed recently, we construct compact finite difference schemes
for the backward fractional Feynman-Kac equation, which has q-th (q=1, 2, 3, 4)
order accuracy in temporal direction and fourth order accuracy in spatial
direction, respectively. In the case q=1, the numerical stability and
convergence of the difference scheme in the discrete L-infinity norm are proved
strictly, where a new inner product is defined for the theoretical analysis.
Finally, numerical examples are provided to verify the effectiveness and
accuracy of the algorithms.
