Fractional Feynman-Kac Equation for Non-Brownian Functionals

We derive backward and forward fractional Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [Phys. Rev. Lett. 96, 230601 (2006)] provide the correct fractional framework for the problem. For applications, we calculate the distribution of occupation times in half space and show how the statistics of anomalous functionals is related to weak ergodicity breaking.

Figure 1

Fluctuations of the time average of $x(t)$, $⟨{\overline{x}}^2⟩$ for a particle in a binding harmonic field, do not vanish when $t\to \infty$ and thus ergodicity is broken. The exception is the case of normal diffusion $\alpha =1$, which exhibits ergodic behavior. The dashed straight lines are the limit $\underset{t\to \infty }{lim}⟨{\overline{x}}^2⟩=(1-\alpha )⟨x^2{⟩}_{\mathrm{th}}$. We use $⟨x^2{⟩}_{\mathrm{th}}=1/2$, $x_0=0$ and hence $⟨{\overline{x}}^2(t=0)⟩=(x_0{)}^2=0$ and according to Eq. (17) for $\alpha <1/3$ $(\alpha >1/3)$ the solution approaches the asymptotic limit from below (above).