论文标题
在线性随机部分 - integro-differential方程的Hölder规律性上
On the Hölder regularity of a linear stochastic partial-integro-differential equation with memory
论文作者
论文摘要
鉴于最新的关于线性粘弹性流体颗粒波动的工作，我们研究了一个线性随机部分 - 元素 - 差异方程，并具有记忆的内存，该方程是由有限的，光滑的域上的固定噪声驱动的。使用〜\ cite {McKinley2018 Anomalous}中引入的广义固定溶液的框架，我们为差分操作员和噪声提供了足够的条件，以获得有关方程的固定溶液的存在以及固定溶液的Hölder规律性。作为规律性结果的应用，我们将随机热方程的类似经典结果进行比较。当1D随机热方程是由白噪声驱动的，解决方案与空间和时间规律性连续，分别是Hölder$（1/2- \ ep）$和$（1/4- \ ep）$。当由彩色空间噪声驱动时，解决方案可以根据噪声结构具有一系列规律性属性。在这里，我们表明，在粘弹性扩散应用中产生的彩色内存的特定形式，满足所谓的波动关系 - 散文关系，产生了hölder$（1/2- \ ep）$（1/2- \ ep）$和$（1/2- \ ep）$的样本路径。
In light of recent work on particles fluctuating in linear viscoelastic fluids, we study a linear stochastic partial-integro-differential equation with memory that is driven by a stationary noise on a bounded, smooth domain. Using the framework of generalized stationary solutions introduced in~\cite{mckinley2018anomalous}, we provide sufficient conditions on the differential operator and the noise to obtain the existence as well as Hölder regularity of the stationary solutions for the concerned equation. As an application of the regularity results, we compare to analogous classical results for the stochastic heat equation. When the 1d stochastic heat equation is driven by white noise, solutions are continuous with space and time regularity that is Hölder $(1/2-\ep)$ and $(1/4-\ep)$ respectively. When driven by colored-in-space noise, solutions can have a range of regularity properties depending on the structure of the noise. Here, we show that the particular form of colored-in-time memory that arises in viscoelastic diffusion applications, satisfying what is called the Fluctuation--Dissipation relationship, yields sample paths that are Hölder $(1/2-\ep)$ and $(1/2-\ep)$ in space and time.
