  Anomalous diffusions are ubiquitous in nature, whose functional distributions
are governed by the backward Feynman-Kac equation. In this paper, the local
discontinuous Galerkin (LDG) method is used to solve the 2D backward
Feynman-Kac equation in a rectangular domain. The spatial semi-discrete LDG
scheme of the equivalent form (obtained by Laplace transform) of the original
equation is established. After discussing the properties of the fractional
substantial calculus, the stability and optimal convergence rates $O(h^{k+1})$
of the semi-discrete scheme are proved by choosing an appropriate generalized
numerical flux. The $L1$ scheme on the graded meshes is used to deal with the
weak singularity of the solution near the initial time. Based on the
theoretical results of a semi-discrete scheme, we investigate the stability and
convergence of the fully discrete scheme, which shows the optimal convergence
rates $O(h^{k+1}+\tau^{\min\{2-\alpha,\gamma\delta\}})$. Numerical experiments
are carried out to show the efficiency and accuracy of the proposed scheme. In
addition, we also verify the effect of the central numerical flux on the
convergence rates and the condition number of the coefficient matrix.
