AI 中文总结
针对非退化Hölder扩散系数的随机热方程，证明当β>2/3时弱唯一性成立，此前同类研究仅在β>3/4时成立，该结果填补了β∈(2/3,3/4]区域内的弱唯一性研究空白。
AI 中文摘要
我们考虑定义在一维环面$\u2124$上的随机热方程（SHE），形式为$\u2202_t u = \u0394 u + g(u)\u1e40W$，其中$\u1e40W$是时空白噪声，$g$是实值函数，满足一致椭圆性（即$|g|$一致远离0有界），且对某个$\u03b2\u2208(0,1)$是全局$\u03b2$-Hölder连续的。我们证明只要$\u03b2>\frac{2}{3}$，弱唯一性成立。对于系数$G$与白噪声维度相同的向量值解，同样的唯一性成立。此前，Mytnik和Perkins（arXiv:0809.0248）通过Yamada-Watanabe论证，在不假设$g$非零的情况下，仅对$\u03b2>\frac{3}{4}$建立了Hölder扩散系数的SHE解的唯一性；当$\u03b2<\frac{3}{4}$时，Mueller、Mytnik和Perkins（arXiv:1201.2767）构造了满足$g(0)=0$的非唯一SPDE例子。后续针对非退化$g$的广义耦合论证也止步于相同的阈值$\frac{3}{4}$。我们的结果表明，$g$的一致椭圆性使SHE在Hölder正则性区域恢复了唯一性，而具有非椭圆$g$和相同Hölder正则性的同类SHE在该区域通常是法律意义上非唯一的。这构成了$\u03b2\u2208(\frac{2}{3},\frac{3}{4}]$区域内第一类SHE弱唯一性的一般性结果。
英文摘要
We consider stochastic heat equation (SHE) defined on 1-d torus $\mathbb{T}$ of the form $$\partial_t u=Δu+g(u)\dot{W},$$where $\dot{W}$ is a space-time white noise and $g$ is a real-valued function which is uniformly elliptic (i.e., $|g|$ is uniformly bounded away from 0), and is globally $β$-Holder continuous for some $β\in(0,1)$. We prove that weak uniqueness holds as long as $β>\frac{2}{3}$. The same uniqueness holds for vector-valued solutions where the coefficient $G$ has the same dimension as the white noise. Previously, uniqueness of solutions to the SHE with Holder diffusion coefficient was only established for $β>\frac{3}{4}$ via a Yamada Watanabe argument by Mytnik and Perkins (arxiv:0809.0248) without assuming $g$ is nonzero. And when $β<\frac{3}{4}$, Mueller, Mytnik and Perkins (arXiv:1201.2767) constructed a non-unique SPDE example satisfying $g(0)=0$. A later generalized coupling argument for nondegenerate $g$ also stopped at the same threshold $\frac{3}{4}$. Our result shows that uniform ellipticity of $g$ restores uniqueness to SHEs in the Holder regime where the same SHE with non-elliptic $g$ and the same Holder regularity are often non-unique in law. This constitutes the first general class of SHE weak uniqueness results in the $β\in(\frac{2}{3},\frac{3}{4}]$ regime.
