论文标题
时间空间白噪声驱动的时间分数随机热方程
The time-fractional stochastic heat equation driven by time-space white noise
论文作者
论文摘要
我们研究了由时间空间白噪声驱动的带有空间尺寸$ d \ in \ mathbb {n} = \ {1,2，... \} $的时空随机热方程，而分数的时间衍生化是（0,2）$ in（0,2）$。我们从分布意义上考虑方程式，并找到了$ \ nathcal {s}'$ - 有价值的解决方案$ y（t，x）$的显式表达式，其中$ \ nathcal {s}'$是perked缩的分布的空间。在Y. Hu \ cite {hu}的术语之后，我们说解决方案是\ emph {py}，如果$ y（t，x）\在l^2（\ mathbb {p}）$ in $ t，x $中，$ t，x $，其中$ \ mathbb {p} $是下一个跨越的时间段的brownian time-space brownian time-space brownian tiers-space brownian sips brownian time-space brownian sipernian。众所周知，在经典情况下，$α= 1 $，仅当空间尺寸$ d = 1 $时，解决方案才是温和的。我们证明，如果$α\在（1,2）$中，如果$ d = 1 $或$ d = 2 $，则解决方案是温和的。如果$α<1 $，我们证明该解决方案对于任何$ d $都不是温和的。
We study the time-fractional stochastic heat equation driven by time-space white noise with space dimension $d\in\mathbb{N}=\{1,2,...\}$ and the fractional time-derivative is the Caputo derivative of order $α\in (0,2)$. We consider the equation in the sense of distribution, and we find an explicit expression for the $\mathcal{S}'$-valued solution $Y(t,x)$, where $\mathcal{S}'$ is the space of tempered distributions. Following the terminology of Y. Hu \cite{Hu}, we say that the solution is \emph{mild} if $Y(t,x) \in L^2(\mathbb{P})$ for all $t,x$, where $\mathbb{P}$ is the probability law of the underlying time-space Brownian motion. It is well-known that in the classical case with $α= 1$, the solution is mild if and only if the space dimension $d=1$. We prove that if $α\in (1,2)$ the solution is mild if $d=1$ or $d=2$. If $α< 1$ we prove that the solution is not mild for any $d$.
