By Matthias Heymann
Presenting a learn of geometric motion functionals (i.e., non-negative functionals at the house of unparameterized orientated rectifiable curves), this monograph makes a speciality of the subclass of these functionals whose neighborhood motion is a degenerate form of Finsler metric that could vanish in yes instructions, making an allowance for curves with optimistic Euclidean size yet with 0 motion. For such functionals, standards are constructed less than which there exists a minimal motion curve top from one given set to a different. Then the homes of this curve are studied, and the non-existence of minimizers is validated in a few settings.
Applied to a geometrical reformulation of the quasipotential of Wentzell-Freidlin conception (a subfield of enormous deviation theory), those effects can yield the life and homes of utmost probability transition curves among metastable states in a stochastic approach with small noise.
The publication assumes simply normal wisdom in graduate-level research; all higher-level mathematical suggestions are brought alongside the way.
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Additional info for Minimum Action Curves in Degenerate Finsler Metrics: Existence and Properties
Sample text
QAA2 with S. Qn / Ä S. n / We then construct a new sequence . Qn /n2N 1 for 8n 2 N as in the proof of part (ii), only that now if xji only has ? weak local minimizers then the curve segment n;i must be obtained from 38 3 Existence of Minimum Action Curves ? xji / in this case. We can assume ? that each segment n;i visits the point xji at most once (otherwise we can cut out the piece between the first and the last hitting point of xji , which can only decrease the action of the curve). 8, to show that some subsequence of the arclength ?
0/ are locally Hölder continuous at x. 9) is fulfilled if and only if x is a critical point. 5. 8. 10) says that every point in the stable and unstable manifold of x (except for x itself) has to lie on a flowline emanating from one of a finite collection of admissible manifolds, or equivalently, that every flowline in the stable and the unstable manifold must intersect one of these finitely many admissible manifolds. See Sect. 4 for examples. 25 (ii) that is responsible for the somewhat excessive length of this monograph (and in particular for Chap.
0; 1/. ˛i / in infinite length. x/ Q ). 0; 1/; or '. 0; 12 /. See Sect. 3 and Fig. 1 for an illustration of these classes of curves. 2) 2 Q we define Z length. 0; 1/ of . 3 Let x 2 D, let the sequence . 25. 0; 1/ ! 0; 12 /. 1. 6 (i). 0; 1/. x/ are defined analogously. 3 Summary of the Various Classes of Curves (See Fig. ) All curves are unparameterized and oriented, and they may have loops and cusps. The class contains only curves with finite length, while curves in Q may reach and/or leave finitely many points in infinite length, also repeatedly.