论文标题
最有可能元素的小额限制是小噪声限制中最有可能的元素
The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit
论文作者
论文摘要
在本文中，我们研究了Onsager-Machlup函数及其与Freidlin-Wentzell函数的关系，以等同于任意无限的高斯高斯度量。 Onsager-Machlup功能可以作为无限尺寸空间的密度，在该空间中不存在均匀度量，并且被视为``最有可能的元素''的拉格朗日。这导致了一个难题 - 这两个功能之间有什么关系？ 我们在freidlin-wentzell函数的小噪声限制下显示了Onsager-Machlup函数的点和$γ$ - convergence（本质上是最小化器的收敛），并给出两者的表达。也就是说，我们表明，最有可能的元素的小噪声极限是无限尺寸测量的小噪声限制中最可能的元素，等于高斯。措施的示例包括依赖路径依赖性随机微分方程的解决方案和无限代数方程的无限系统的定律。
In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the ``most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the ``most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and $Γ$-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations.
