  I study a class of action functionals on the space of unparameterized
oriented rectifiable curves in R^n. The local action is a degenerate type of
Finsler metric that may vanish in certain directions, thus allowing for curves
with positive Euclidean length but zero action. Given two sets A_1 and A_2, I
develop criteria under which there exists a minimum action curve leading from
A_1 to A_2. I then study the properties of these minimizers, and I prove the
non-existence of minimizers in some situations. Applied to a geometric
reformulation of the quasipotential of large deviation theory, my results can
prove the existence and properties of maximum likelihood transition curves
between two metastable states in a stochastic process with small noise.
