多重齐次测度与随机极坐标表示
Multihomogeneous Measures and Stochastic Polar Representations
- University of Lausanne（洛桑大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究多重齐次测度的随机极坐标表示，推导了其随机积分表示的等价条件，刻画了同测度的随机元特征及平稳性，还构造了半正定尾重叠核。
AI中文摘要：
设$q\in\mathbb N$，$G=(0,\infty)^q$，$S:G\times E\longrightarrow E, (r,x)\longmapsto S_rx$是任意可测空间$(E,\mathcal E)$上的联合可测左作用。对$\alpha=(\alpha_1,\ldots,\alpha_q)\in(0,\infty)^q$，定义$\chi_\alpha(r)=\prod_{i=1}^q r_i^{\alpha_i}$。本文研究满足$\nu(S_rA)=\chi_\alpha(r)^{-1}\nu(A), r\in G, A\in\mathcal E$的非零$\sigma$-有限测度$\nu$。受文献[1]中标量情形$q=1$的启发，我们推导了存在$E$值随机元$Z$使得$\nu(A) = \mathbb{E}\{\int_G\mathbb I_A(S_rZ)\prod_{i=1}^q \alpha_i r_i^{-\alpha_i-1}dr_i\}, A\in\mathcal E$的等价条件。我们还利用多重齐次矩，以及在固定容许乘积规范后的加权横向测度，刻画了两个随机元生成同一齐次测度的条件。当相应加权横向测度有限时，倾斜与规范归一化可得到一个典型表示元，其在指定规范壳上的分布唯一。最后，我们刻画了在与$S$交换的作用下的平稳性，并直接由$\nu$构造了半正定尾重叠核。
英文摘要：
Let \(q\in\mathbb N\), let \(G=(0,\infty)^q\), and let $ S:G\times E\longrightarrow E, (r,x)\longmapsto S_rx $ be a jointly measurable left action on an arbitrary measurable space \((E,\mathcal E)\). For \(α=(α_1,\ldots,α_q)\in(0,\infty)^q\) set $ χ_α(r)=\prod_{i=1}^q r_i^{α_i}. $ We study nonzero \(σ\)-finite measures \(ν\) satisfying $ ν(S_rA)=χ_α(r)^{-1}ν(A), r\in G, A\in\mathcal E.$ Motivated by the scalar case \(q=1\) studied in [1] we derive equivalent conditions for the existence of an \(E\)-valued random element \(Z\) such that $ ν(A) = \mathbb{E}\{\int_G\mathbb I_A(S_rZ)\prod_{i=1}^q α_i r_i^{-α_i-1}dr_i\}, A\in\mathcal E. $ We also characterise when two random elements generate the same homogeneous measure, using multihomogeneous moments and, after fixing an admissible product gauge, weighted transverse measures. When the corresponding weighted transverse measure is finite, tilting and gauge normalisation produce a canonical representer, unique in law on the prescribed gauge shell. Finally, we characterise stationarity under an action commuting with \(S\) and construct positive semidefinite tail-overlap kernels directly from \(ν\).
