时间依赖分数噪声下最可能路径的持久性与长时间破裂及其在KAM环面中的应用

中文摘要 AI 辅助

我们通过Onsager-Machlup泛函研究由分数布朗运动驱动的多维随机微分方程中最可能路径的持久性，其中扩散系数依赖于时间，Hurst参数$H\in(1/4,1)$。在适当的结构性和变分条件下，对于足够小的噪声，确定性轨迹在固定端点转移问题和自由端点演化问题中均保持为最可能路径，而足够大的噪声则破坏其局部极小性。更一般地，当精确持久性不成立时，全局最可能路径在一致拓扑和Hölder拓扑下均以$O(\epsilon)$的速率收敛到无噪声系统的相应轨迹。我们进一步分析了长度为$NT$的时间区间上周期确定性轨迹的二阶变分。对于$H>1/2$，在足够长的区间上正定性（从而局部极小性）会丧失。对于$H\in(1/4,1/2]$，固定端点问题的长时间正定性成立，但该结论不能直接推广到自由端点情形。我们还建立了近可积哈密顿系统中KAM环面在最可能演化路径意义下的持久性。最后，一个二维数值例子说明了小噪声下确定性轨迹的持久性以及大噪声下它们的显著偏离。

英文摘要

We investigate the persistence of most probable paths through the Onsager--Machlup functional for multidimensional stochastic differential equations driven by fractional Brownian motion with time-dependent diffusion coefficients and Hurst parameter $H\in(1/4,1)$. Under suitable structural and variational conditions, deterministic trajectories remain most probable paths for sufficiently small noise in both the fixed-endpoint transition problem and the free-endpoint evolution problem, whereas sufficiently large noise destroys their local minimality. More generally, when exact persistence does not hold, global most probable paths converge to the corresponding trajectories of the noise-free system in both the uniform and Hölder topologies at the rate $O(ε)$. We further analyze the second variation along periodic deterministic trajectories over time intervals of length $NT$. For $H>1/2$, positive definiteness, and hence local minimality, is lost on sufficiently long intervals. For $H\in(1/4,1/2]$, long-time positive definiteness holds for the fixed-endpoint problem, but this conclusion does not directly extend to the free-endpoint setting. We also establish the persistence of KAM tori in nearly integrable Hamiltonian systems in the sense of most probable evolution paths. Finally, a two-dimensional numerical example illustrates the persistence of deterministic trajectories under small noise and their pronounced deviation under large noise.

发表机构

• College of Mathematics, Jilin University（吉林大学数学学院）

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