论文标题
一类新的分数抛物线随机演化方程的规律性理论
Regularity theory for a new class of fractional parabolic stochastic evolution equations
论文作者
论文摘要
形式的$（partial_t + a）^γx（t）= \ dot {w}^q（t）$，$ t \ in [0，t] $，$γ\ in（in（0，\ infty）$ s $ c _ $ s $ h hill hill h y a a a a $ hill hill，时空驱动噪声$ \ dot {w}^q $是$ h $ valued canlindrical $ q $ weener流程的正式时间派生。定义了轻度和弱解决方案；这些概念被证明是等效的，并导致了良好的问题。研究了解决方案过程的时间和空间规律性$ x $，前者通过均方或路径平滑度来衡量，并通过使用$ a $的分数幂域来衡量。此外，分析了$ x $的协方差及其长期行为。 这些抽象的结果应用于$ a：= l^β$和$ q：= \ tilde {l}^{ - α} $的情况，是对称，强烈椭圆形的二阶差异操作员（I）有界的欧几里得域或（II）平滑，平滑的平滑表面的对称，强烈椭圆形的二阶差分运算符的分数。在这些情况下，高斯溶液过程可以看作是仅空间（惠特 - ）Matérn领域到时空的概括。
A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^γX(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $γ\in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert space $H$ and the spatiotemporal driving noise $\dot{W}^Q$ is the formal time derivative of an $H$-valued cylindrical $Q$-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process $X$ are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of $A$. In addition, the covariance of $X$ and its long-time behavior are analyzed. These abstract results are applied to the cases when $A := L^β$ and $Q:=\tilde{L}^{-α}$ are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle-)Matérn fields to space-time.
