W-马尔可夫测度、转移算子、小波和多分辨
W-Markov measures, transfer operators, wavelets and multiresolutions
摘要：
在一般情况下，我们解决了以下逆问题：给定一个正算子$R$，作用于固定测度空间$(X,\mathcal B_X)$上的可测函数，我们构造了一个相关的马尔可夫链。具体地说，从选择$R$（转移算子）和$(X, \mathcal B_X)$上的概率测度$μ_0$开始，然后我们建立一个相关的马尔可夫链$T_0, T_1, T_2,\ldots$，这些随机变量（r.v）在一个合适的概率空间$(Ω,\mathcal F, \mathbb P)$中实现，每个r. v在$X$中取值，$T_0$以概率$μ_0$为定律。我们进一步展示了$R$的光谱数据，例如，$R$ -调和函数的存在，传播到马尔可夫链。相反，在一般情况下，我们表明，每一个马尔可夫链是由它的转移算子。在一系列的例子中，我们把这种对应关系转化为实际的术语：$(i)$迭代函数系统（IFS），$(ii)$小波多分辨率结构，和$(iii)$ IFS随机控制。
英文摘要：
In a general setting we solve the following inverse problem: Given a positive operators $R$, acting on measurable functions on a fixed measure space $(X,\mathcal B_X)$, we construct an associated Markov chain. Specifically, starting with a choice of $R$ (the transfer operator), and a probability measure $μ_0$ on $(X, \mathcal B_X)$, we then build an associated Markov chain $T_0, T_1, T_2,\ldots$, with these random variables (r.v) realized in a suitable probability space $(Ω,\mathcal F, \mathbb P)$, and each r.v. taking values in $X$, and with $T_0$ having the probability $μ_0$ as law. We further show how spectral data for $R$, e.g., the presence of $R$-harmonic functions, propagate to the Markov chain. Conversely, in a general setting, we show that every Markov chain is determined by its transfer operator. In a range of examples we put this correspondence into practical terms: $(i)$ iterated function systems (IFS), $(ii)$ wavelet multiresolution constructions, and $(iii)$ IFSs with random control.
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。
