论文标题
具有加性噪声的SPDE的最佳规律性
Optimal regularity of SPDEs with additive noise
论文作者
论文摘要
随机场溶液对随机部分微分方程（SPDE）的样品 - 功能规律性自然取决于外部噪声的粗糙度，以及用于定义方程式的基础内部差异算子的属性。在本文中，我们考虑$ 0，\ infty）\ times \ times \ mathbb {r}^d $的抛物面和双曲线，$ \ partial_t u = l u = l u = l u + g（u） + \ dot {f} f} f}数据，用时空均匀的高斯噪声$ \ dot {f} $在其时间变量中是白色，并且在其太空变量中相关，并且由真正的$ d $ d $ d $ d $ d $ d $lévy流程$ x $驱动。我们发现针对这些SPDE的各个随机场溶液的最佳Hölder条件。我们的条件是根据索引描述了该过程$ x $特征指数某些功能的集成性的索引，相对于$ \ dot f $的空间协方差的光谱度量。这些指标是由参考文献[45，46]提出的，因为特定情况是$ l $是$ \ mathbb {r}^d $上的laplace运营商。
The sample-function regularity of the random-field solution to a stochastic partial differential equation (SPDE) depends naturally on the roughness of the external noise, as well as on the properties of the underlying integro-differential operator that is used to define the equation. In this paper, we consider parabolic and hyperbolic SPDEs on $0,\infty)\times\mathbb{R}^d$ of the form $\partial_t u = L u + g(u) + \dot{F} \qquad\text{and}\qquad \partial^2_t u = L u + c + \dot{F}, $ with suitable initial data, forced with a space-time homogeneous Gaussian noise $\dot{F}$ that is white in its time variable and correlated in its space variable, and driven by the generator $L$ of a genuinely $d$-dimensional Lévy process $X$. We find optimal Hölder conditions for the respective random-field solutions to these SPDEs. Our conditions are stated in terms of indices that describe thresholds on the integrability of some functionals of the characteristic exponent of the process $X$ with respect to the spectral measure of the spatial covariance of $\dot F$. Those indices are suggested by references [45, 46] on the particular case that $L$ is the Laplace operator on $\mathbb{R}^d$.
