Existence and Properties of Minimum Action Curves for Degenerate Finsler Metrics

TL;DR

This paper investigates the existence and characteristics of minimum action curves within a class of degenerate Finsler metrics, with applications to large deviation theory and stochastic processes.

Contribution

It establishes criteria for the existence of minimum action curves in degenerate Finsler metrics and analyzes their properties, including non-existence in certain cases.

Findings

Criteria for existence of minimum action curves

Properties of minimizers in degenerate Finsler metrics

Non-existence results in specific scenarios

Abstract

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.