  Functionals of particles' paths have diverse applications in physics,
mathematics, hydrology, economics, and other fields. Under the framework of
continuous time random walk (CTRW), the governing equations for the probability
density functions (PDFs) of the functionals, including the ones of the paths of
stochastic processes of normal diffusion, anomalous diffusion, and even the
diffusion with reaction, have been derived. Sometimes, the stochastic processes
in physics and chemistry are naturally described by Langevin equations. The
Langevin picture has the advantages in studying the dynamics with an external
force field and analyzing the effect of noise resulting from a fluctuating
environment. We derive the governing equations of the PDFs of the functionals
of paths of Langevin system with both space and time dependent force field and
arbitrary multiplicative noise; and the backward version is proposed for the
system with arbitrary additive noise or multiplicative Gaussian white noise
together with a force field. For the newly built equations, their applications
of solving the PDFs of the occupation time and area under the trajectory curve
are provided, and the results are confirmed by simulations.
