AI 中文总结
研究由独立性保持产生的概率律特征化问题，引入平面网几何，通过实解析微分同胚\(F\)研究独立性保持条件，为相关定理提供统一证明，给出解空间性质，还产生更多例子，有望实现保持三参数族测度独立性映射的完全分类。
AI 中文摘要
本文将平面网几何引入到由独立性保持产生的概率律的特征化问题研究中。设\(F(x,y)=(u,v)\)是定义在\(\mathbb{R}^2\)中一个开域上的实解析微分同胚。设\(X\)和\(Y\)是独立随机变量，令\((U,V)=F(X,Y)\)。若\(U\)和\(V\)独立，则称映射\(F\)保持独立性。该条件具有高度限制性，可用于刻画满足此条件的概率律。此框架为特征化定理提供了统一证明，涵盖经典结果如Kac - Bernstein和Lukacs定理以及Matsumoto - Yor性质，还有关于四有理Yang - Baxter映射\(H_I^+\)、\(H_{II}^+\)和\(H_{III}^A\)的近期结果。对于具有正连续密度的测度，当且仅当其对数密度解相应的非齐次阿贝尔泛函方程时，映射保持独立性。解空间与一个以阿贝尔关系空间为模型的仿射空间等同，根据Bol秩界其维数至多为三。同样的方法还产生了更多例子，包括一类推广Koudou和Vallois所研究变换的例子；这类中的每个映射都至少有一个双参数族的保持乘积测度。映射的独立性保持性质在坐标方向的单变量重参数化下保持不变，这也保持了相关的网。因此，与映射相关的网是在独立性保持映射的相应等价关系下的自然不变量。结合最大秩的非奇异平面\(4 -\)网的局部代数性，这种对应关系为保持三参数族测度独立性的映射的完全分类提供了一条途径。
英文摘要
This paper introduces planar web geometry into the study of characterization problems for probability laws arising from independence preservation. Let $F(x,y) = (u, v)$ be a real-analytic diffeomorphism defined on an open domain in $\mathbb{R}^2$. Let $X$ and $Y$ be independent random variables and set $(U, V) = F(X, Y)$. The map $F$ is said to preserve the independence if $U$ and $V$ are independent. This condition is highly restrictive and can therefore be used to characterize the laws for which it holds. This framework provides unified proofs of characterization theorems, including classical results such as the Kac--Bernstein and Lukacs theorems and the Matsumoto--Yor property, as well as recent results for the quadrirational Yang--Baxter maps $H_I^+$, $H_{II}^+$, and $H_{III}^A$. For measures with positive continuous densities, the map preserves independence if and only if their logarithmic densities solve the corresponding inhomogeneous Abelian functional equation. The solution space is identified with an affine space modeled on the space of Abelian relations, whose dimension is at most three according to Bol's rank bound. The same method also produces further examples, including a class generalizing transformations studied by Koudou and Vallois; every map in this class admits at least a two-parameter family of preserved product measures. The independence-preserving property of a map is preserved by coordinatewise one-variable reparametrizations, which also preserve the associated web. Thus the web associated with a map is a natural invariant under the corresponding equivalence relation of independence-preserving maps. Together with the local algebraizability of nonsingular planar $4$-webs of maximal rank, this correspondence suggests a route toward a complete classification of maps preserving the independence of a three-parameter family of measures.
Comments9 pages
