FeynmanKac equation for anomalous processes with space and timedependent forces
Abstract
Functionals of a stochastic process Y(t) model many physical timeextensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y(t) is a normal diffusive process, their statistics are obtained as the solution of the celebrated FeynmanKac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic secondorder partial differential equations. When Y(t) is nonBrownian, e.g. an anomalous diffusive process, generalizations of the FeynmanKac equation that incorporate powerlaw or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Lévy process whose Laplace exponent is directly related to the memory kernel appearing in the generalized FeynmanKac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in numerous experiments on biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both space and timedependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the FeynmanKac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are of particular relevance for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple nonequilibrium model are obtained. An additional aim of this paper is to provide a pedagogical introduction to Lévy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 April 2017
 DOI:
 10.1088/17518121/aa5a97
 arXiv:
 arXiv:1701.01641
 Bibcode:
 2017JPhA...50p4002C
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Mathematical Physics
 EPrint:
 Invited contribution to the J. Phys. A special issue Emerging Talents