一类基于Sobolev空间建模的非参数统计流形
A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space
AI总结：
本文构造了一族基于加权Sobolev空间建模的有限测度非参数无限维流形，支持Fisher-Rao弱黎曼度量，利用变形指数函数与Nemytskii算子保证连续性，并概述了其在非线性滤波随机偏微分方程中的应用。
摘要：
我们构造了一族 $R^d$ 上有限测度的非参数（无限维）流形。这些流形以多种加权Sobolev空间为模型，包括Hilbert-Sobolev空间和混合范数空间。每个流形都支持Fisher-Rao度量作为弱黎曼度量。密度通过具有线性增长的变形指数函数来表示。在Sobolev背景下不同寻常的是，作为其线性增长的结果，这“提升”为一个非线性叠加（Nemytskii）算子，该算子在一类特定的混合范数模型空间以及固定范数空间 $W^{2,1}$ 上连续作用；即它将这些空间各自连续映射到自身。它还在其他固定范数空间之间连续映射，伴随着随导数数量增加而增大的Lebesgue指数损失。部分结果实质性地利用了log-Sobolev嵌入定理。每个流形都包含一个概率测度的光滑嵌入子流形。文中概述了其在非线性滤波（进而Fokker-Planck方程）的随机偏微分方程中的应用。
英文摘要：
We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on $R^d$. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space $W^{2,1}$; i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
