  We prove pathwise large-deviation principles of switching Markov processes by
exploiting the connection to associated Hamilton-Jacobi equations, following
Jin Feng's and Thomas Kurtz's method. In the limit that we consider, we show
how the large-deviation problem in path-space reduces to a spectral problem of
finding principal eigenvalues. The large-deviation rate functions are given in
action-integral form.
  As an application, we demonstrate how macroscopic transport properties of
stochastic models of molecular motors can be deduced from an associated
principal-eigenvalue problem. The precise characterization of the macroscopic
velocity in terms of principal eigenvalues implies that breaking of detailed
balance is necessary for obtaining transport. In this way, we extend and unify
existing results about molecular motors and place them in the framework of
stochastic processes and large-deviation theory.
