刘维尔布朗运动的响应演算 I：简单谱、联合本征值密度与沃德恒等式
Response Calculus for Spectral Simplicity and Joint Eigenvalue Densities
浏览论文内容
中文总结 AI 辅助
该研究针对狄利克雷刘维尔布朗运动建立响应演算，证明其生成元在0<γ<2内几乎必然有简单谱，得到预解式可微性及沃德恒等式，解决了相关公开问题。
中文摘要 AI 辅助
我们针对高斯自由场的Cameron–Martin平移下的狄利克雷刘维尔布朗运动建立了一套响应演算。在每个有界连通平面域上，无需边界正则性假设，我们在整个次临界范围0<γ<2内证明：生成元几乎必然具有简单谱，且任意有限个有序本征值构成的向量具有绝对连续分布，这解决了狄利克雷刘维尔布朗运动的简单谱公开问题。该演算结合了高斯乘性混沌的相干版本与相关迹形式的固定空间扰动：孤立本征值簇的一阶变分在其本征空间上成为有限维压缩；每个多重本征值簇存在具有简单一阶分裂的光滑方向，而不同简单本征值的响应测度线性无关。有限维高斯切片上的解析零集与淹没论证进而给出谱结论。在算子层面，于同一整个次临界范围，该演算给出预解式在固定能量空间中的可微性，并验证了单边、移动测度型及固定基型预解式沃德恒等式。我们构造了显式因果缓增分布，其拉普拉斯变换实现移动测度型与固定基型响应；未断言时域热半群差商收敛。最后，对0<γ<√2，空间平均预解式观测值的响应测度的Green–Riesz位势实现其高斯Sobolev梯度，这在所有参数下给出占据预解式的绝对连续性，且在预解式参数的确定性零集之外，对进一步的标量观测值及与不交非负测试函数相关的有限族也成立。
英文摘要
We develop a perturbative response calculus for spectral problems obtained by changing the speed measure of a fixed symmetric energy form in a Gaussian environment. If Cameron--Martin translation acts by $μ_{x+f}=e^{κf}μ_x$, unitary transport identifies the varying $L^2$ spaces and produces a common-domain analytic family. For a positive eigenvalue $Λ$, the first-order operator is $-κΛ$ times the compression of multiplication by $f$ to the $Λ$-eigenspace; its eigenvalues are the derivatives of the analytic branches issuing from $Λ$. A countable separation condition and finite-dimensional Gaussian disintegration then give almost-sure simplicity; a response-transversality condition and the inverse function theorem give joint densities for all finite vectors of positive ordered eigenvalues. We verify these hypotheses for every $0<γ<2$ in two models: Dirichlet Liouville Brownian motion on an arbitrary bounded connected planar domain, and the Liouville--Cauchy operator on the circle. In the first model the whole spectrum is almost surely simple; in the second the constants form the deterministic zero mode and the positive spectrum is almost surely simple. Transversality follows from a local eigenfunction-square identity in the Brownian case and its nonlocal jump-form analogue in the Cauchy case.
