We give a rigorous proof of the equivalence of Manes supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak...We give a rigorous proof of the equivalence of Manes supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather＇s a- function is discussed.展开更多
The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise.However,it suffers from a notorious issue that the functional is unb...The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise.However,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to infinity.This hinders the interpretation of the results obtained by minimizing the OM functional.We provide a new perspective on this issue.Under mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of curves.The graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal energy.We further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path simultaneously.This algorithm is successfully applied to several typical examples in rare event studies.Some interesting comparisons with the Freidlin-Wentzell action functional are also made.展开更多
基金Supported by the National Basic Research Program of China (Grant No. 2007CB814800)Natural Scientific Foundation of China (Grant No. 10971093)
文摘We give a rigorous proof of the equivalence of Manes supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather＇s a- function is discussed.
基金supported by National Natural Science Foundation of China(Grant Nos.11421101,91530322 and 11825102)supported by the Construct Program of the Key Discipline in Hunan Province+1 种基金supported by Singapore Ministry of Education Academic Research Funds(Grant Nos.R-146-000-267-114 and R-146-000-232-112)National Natural Science Foundation of China(Grant No.11871365)。
文摘The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing noise.However,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to infinity.This hinders the interpretation of the results obtained by minimizing the OM functional.We provide a new perspective on this issue.Under mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of curves.The graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal energy.We further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path simultaneously.This algorithm is successfully applied to several typical examples in rare event studies.Some interesting comparisons with the Freidlin-Wentzell action functional are also made.
