  This work is devoted to deriving the Onsager-Machlup action functional for a
class of stochastic differential equations with (non-Gaussian) L\'{e}vy process
as well as Brownian motion in high dimensions. This is achieved by applying the
Girsanov transformation for probability measures and then by a path
representation. The Poincar\'{e} lemma is essential to handle such path
representation problem in high dimensions. We provide a sufficient condition on
the vector field such that this path representation holds in high dimensions.
Moreover, this Onsager-Machlup action functional may be considered as the
integral of a Lagrangian. Finally, by a variational principle, we investigate
the most probable transition pathways analytically and numerically.
