  For a stochastic system, its evolution from one state to another can have a
large number of possible paths. Non-uniformity in the field of system variables
leads the local dynamics in state transition varies considerably from path to
path and thus the distribution of the paths affects statistical characteristics
of the system. Such a characteristic can be referred to as path-dependence of a
system, and long-time correlation is an intrinsic feature of path-dependence
systems. We employed a local path density operator to describe the distribution
of state transition paths, and based on which we derived a new kinetic equation
for path-dependent systems. The kinetic equation is similar in form to the
Kramers-Moyal expansion, but with its expansion coefficients determined by the
cumulants with respect to state transition paths, instead of transition
moments. This characteristic makes it capable of accounting for the non-local
feature of systems, which is essential in studies of large scale systems where
the path-dependence is prominent. Short-time correlation approximation is also
discussed. It shows that the cumulants of state transition paths are equivalent
to jump moments when correlation time scales are infinitesimal, as makes the
kinetic equation derived in this paper has the same physical consideration of
the Fokker-Planck equation for Markov processes.
