Existence and Properties of Minimum Action Curves for Degenerate Finsler Metrics

Details Existence and Properties of Minimum Action Curves for Degenerate Finsler Metrics Existence and Properties of Minimum Action Curves for Degenerate Finsler Metrics

2010-04-27

arxiv.org/abs/1004.4873v1

Mathematics

Probability

Scientific paper

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.

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