退化Finsler算子最小作用曲线的存在性及性质
Existence and Properties of Minimum Action Curves for Degenerate Finsler Metrics
摘要：
本文研究了R^n中非参数定向可求长曲线空间上的一类作用泛函。局部作用量是芬斯勒度量的一种退化类型，在某些方向上可能为零，因此允许曲线具有正的欧几里德长度但零作用量。给定两个集合A_1和A_2，我给出了存在从A_1到A_2的最小作用曲线的判据。然后，我研究这些极小的属性，我证明了不存在极小在某些情况下。将所得结果应用于大偏差理论的拟势的几何变换，可以证明小噪声随机过程中两个亚稳态之间极大似然过渡曲线的存在性及其性质.
英文摘要：
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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