论文标题
随机热方程式使用Hölder系数的指数千古性
Exponential ergodicity of stochastic heat equations with Hölder coefficients
论文作者
论文摘要
我们研究了由抽象希尔伯特空间上定义的时空白噪声驱动的随机热方程，假设漂移和扩散系数都只是Hölder连续的。随机场SPDE被涵盖为特殊示例。我们给出了第一个证明，即当扩散系数为$β$-Hölder连续$β> \ frac {3} {4} {4} $且均匀非分化时，存在于法律中的独特解决方案。同时，假设SPDE存在合适的Lyapunov函数，我们证明该解决方案相对于典型的Wasserstein距离，该解决方案将指数迅速收敛到独特的不变度。当SPDE具有汉堡类型的非线性$（ - a）^{\ vartheta} f（x_t）$的汉堡类型时，我们的技术概括了（0,1）$，其中$ f $ as $ f $ as $ \ vartheta+vartheta+ε$ - hölder-hölder持续且具有线性增长。对于$ \ vartheta \ in（\ frac {1} {2}，1）$，即使在加法噪声的情况下，此结果也是新结果。
We investigate the stochastic heat equation driven by space-time white noise defined on an abstract Hilbert space, assuming that the drift and diffusion coefficients are both merely Hölder continuous. Random field SPDEs are covered as special examples. We give the first proof that there exists a unique in law mild solution when the diffusion coefficient is $β$ - Hölder continuous for $β>\frac{3}{4}$ and uniformly non-degenerate, and that the drift is locally Hölder continuous. Meanwhile, assuming the existence of a suitable Lyapunov function for the SPDE, we prove that the solution converges exponentially fast to the unique invariant measure with respect to a typical Wasserstein distance. Our technique generalizes when the SPDE has a Burgers type non-linearity $(-A)^{\vartheta}F(X_t)$ for any $\vartheta\in(0,1)$, where $F$ is $\vartheta+ε$- Hölder continuous and has linear growth. For $\vartheta\in(\frac{1}{2},1)$ this result is new even in the case of additive noise.
