Abstract
In this chapter we begin by teaching the reader all the necessary basics of rectifiable curves and absolutely continuous functions. We then introduce the class of geometric action functionals to which our theory can be applied (and in particular the subclass of Hamiltonian geometric actions), give several examples of geometric actions, and prove a lower semi-continuity property for them. Finally, we define the notion of a “drift” of an action, as a generalization of the drift vector field entering the Wentzell-Freidlin action.
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Notes
- 1.
The key argument for this can be found at the end of the proof of Proposition 3.25.
- 2.
This work also proposed an algorithm, called the geometric minimum action method (gMAM ), for numerically computing minimizing curves of such geometric actions.
- 3.
At the beginning of [10], additional smoothness assumptions on H were made, but they do not enter the proofs of these representations.
- 4.
Probabilists will find some comments in Appendix A.6.
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Heymann, M. (2015). Geometric Action Functionals. In: Minimum Action Curves in Degenerate Finsler Metrics. Lecture Notes in Mathematics, vol 2134. Springer, Cham. https://doi.org/10.1007/978-3-319-17753-3_2
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