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Functionals of a stochastic process Y(t) model many physical time-extensiveobservables, e.g. particle positions, local and occupation times or accumulatedmechanical work. When Y(t) is a normal diffusive process, their statistics areobtained as the solution of the Feynman-Kac equation. This equation providesthe crucial link between the expected values of diffusion processes and thesolutions of deterministic second-order partial differential equations. WhenY(t) is an anomalous diffusive process, generalizations of the Feynman-Kacequation that incorporate power-law or more general waiting time distributionsof the underlying random walk have recently been derived. A generalrepresentation of such waiting times is provided in terms of a L\'evy processwhose Laplace exponent is related to the memory kernel appearing in thegeneralized Feynman-Kac equation. The corresponding anomalous processes havebeen shown to capture nonlinear mean square displacements exhibiting crossoversbetween different scaling regimes, which have been observed in biologicalsystems like migrating cells or diffusing macromolecules in intracellularenvironments. However, the case where both space- and time-dependent forcesdrive the dynamics of the generalized anomalous process has not been solvedyet. Here, we present the missing derivation of the Feynman-Kac equation insuch general case by using the subordination technique. Furthermore, we discussits extension to functionals explicitly depending on time, which are relevantfor the stochastic thermodynamics of anomalous diffusive systems. Exact resultson the work fluctuations of a simple non-equilibrium model are obtained. Inthis paper we also provide a pedagogical introduction to L\'evy processes,semimartingales and their associated stochastic calculus, which underlie themathematical formulation of anomalous diffusion as a subordinated process.

Feynman-Kac equation for anomalous processes with space- and time-dependent forces