非高斯布朗时间变换的谱简单性与联合特征值密度
Spectral Simplicity and Joint Eigenvalue Densities for a Non-Gaussian Brownian Time Change
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中文总结 AI 辅助
该研究针对单位正方形上带独立非高斯系数狄利克雷特征函数展开构造的速度测度的布朗时间变换，通过鞅论证、酉变换等方法证明其谱简单性并得到联合特征值密度。
中文摘要 AI 辅助
我们研究单位正方形上的布朗时间变换，其速度测度由带独立非高斯系数的狄利克雷特征函数展开构造。对于0<γ<√2，该测度通过二阶矩鞅论证得到。有限系数平移会引发速度测度的相干指数倾斜，对互补系数取条件时，其在每个有限维轨道上给出正勒贝格密度。沿这些轨道的酉变换生成公共域解析族与显式一阶簇导数，一阶分裂论证证明几乎必然简单性，而局部特征函数平方恒等式与范德蒙德论证给出所有有序正特征值有限向量的联合密度。
英文摘要
We study a Brownian time change on the unit square whose speed measure is constructed from a Dirichlet eigenfunction expansion with independent non-Gaussian coefficients. For $0<γ<\sqrt2$, the measure is obtained by a second-moment martingale argument. Finite coefficient translations induce coherent exponential tilts of the speed measure, and conditioning on the complementary coefficients gives positive Lebesgue densities on every finite-dimensional orbit. Unitary transport along these orbits gives a common-domain analytic family and explicit first-order cluster derivatives. A first-order splitting argument proves almost-sure simplicity, while the local eigenfunction-square identity and a Vandermonde argument give joint densities for all finite vectors of ordered positive eigenvalues.
