发表机构
Institute for Advanced Algorithms Research; Peking University（高级算法研究所; 北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究扩散边际下抛物方程的加权 $L^2$ Hessian 估计，提出最优大阻尼常数公式，并证明在一般扩散与系数失配下估计的有限性及常数界。
AI 中文摘要
我们研究在扩散边际下的加权 $L^2$ Hessian 估计，允许方程的主矩阵被独立选择。对于正则一致椭圆系数、有界参考漂移和紧支撑加倍初始分布，全区间上的最优大阻尼常数是初始、平坦和传播贡献的最大值。有限性给出加权 Sobolev 适定性和非线性扰动的收缩准则。一个显式算子公式确定初始贡献；在均匀方向极限或无穷远渐近各向同性下，传播界变得精确。在包括非负 Ricci 曲率的几何假设下，传播由作用最小化路径的末端动量决定。对于参考协方差 $a$，匹配主矩阵 $a/2$ 和 Hessian 归一化 $a^{1/2}D^2u\\,a^{1/2}$，每个初始分布都给出有限估计。在没有额外时间权重的情况下，极限常数介于 $2$ 和 $2\sqrt2$ 之间：有限原子分布达到上界，而非退化高斯分布及其有限混合达到下界。
英文摘要
We study weighted $L^2$ Hessian estimates for parabolic terminal equations under the marginals of a prescribed diffusion, allowing singular initial laws and different principal coefficients in the equation and the diffusion. Under coefficient regularity, uniform ellipticity and bounded reference drift, we characterize the limiting optimal constant uniformly over shrinking subintervals with zero terminal data. Its finiteness is equivalent to linear well-posedness in the weighted parabolic Sobolev space on the full interval. For initial laws satisfying weighted doubling and coercivity conditions, this constant is the maximum of contributions from heat evolutions of rescaled initial measures, constant weights and exponential mixtures. The phase contributions are bounded above in terms of the coefficients at time zero outside the initial support and at spatial infinity. Under radial reference structure, we obtain exact coefficient formulas invariant under bounded changes of reference drift. In the matching case, with reference covariance $a$, principal matrix $a/2$ and normalized Hessian $a^{1/2}D^2u\,a^{1/2}$, the constant is $2$ for every initial law with time weight $t^α$, $α\ge1/2$; point starts attain $2\sqrt2$ at $α=0$. Whenever the limiting constant is finite, increasing damping makes the full-interval Hessian constant converge to it and the value and gradient gains tend to zero. As applications, we obtain nonlinear solvability and residual stability for small Hessian perturbations. For solutions and candidates with a common regularity bound, secant estimates give linear stability under a relative spectral gap, while covariance interpolation extends residual stability beyond it.
