Fractional Feynman-Kac Equation with Space-Dependent Anomalous Exponent

To describe the distribution of functionals for inhomogeneous subdiffusion in space- and time-dependent force field, we derive forward and backward time-fractional Feynman-Kac equations with space-dependent anomalous exponent based on the space-jump random walk model. In our examples, we get the statistic of occupation fraction and first passage time for anomalous infiltration of free particles in disordered systems by applying the backward version.