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I study a class of action functionals on the space of unparameterizedoriented rectifiable curves in R^n. The local action is a degenerate type ofFinsler metric that may vanish in certain directions, thus allowing for curveswith positive Euclidean length but zero action. Given two sets A_1 and A_2, Idevelop criteria under which there exists a minimum action curve leading fromA_1 to A_2. I then study the properties of these minimizers, and I prove thenon-existence of minimizers in some situations. Applied to a geometricreformulation of the quasipotential of large deviation theory, my results canprove the existence and properties of maximum likelihood transition curvesbetween two metastable states in a stochastic process with small noise.

Existence and Properties of Minimum Action Curves for Degenerate Finsler Metrics