随机分数阶输运热方程
The stochastic fractional transport heat equation
AI总结：
本文研究分数阶时间随机热方程，分加性与乘性噪声两种情形，给出显式解或卷积方程与Wiener混沌展开，并探讨端点α=2的随机波动方程及温和解条件。
AI中文摘要：
我们介绍了由1+d参数分数阶时间-空间白噪声驱动的分数阶时间随机热方程，分以下两种情况：（i）加性分数阶时间-空间白噪声；（ii）乘性时间-空间布朗噪声。分数阶时间导数被解释为阶数$\alpha \in (0,2]$的Caputo导数，我们假设时间-空间分数阶白噪声的Hurst系数$H=(H_0,H_1,H_2,...,H_d)$位于$(\tfrac{1}{2},1)^{1+d}$中。在加性噪声情形（i）中，我们找到了方程在分布意义下唯一解的显式表达式。在乘性情形（ii）中，我们推导了正确的时空卷积方程及其Wiener混沌展开。对于$1<\alpha\leq2$，需要第二个初始条件。端点$\alpha=2$被单独处理，并产生具有有限传播速度的随机波动方程。解$Y(t,x)$被称为温和解，如果对所有$t,x$有$E[Y^2(t,x)] < \infty$。在乘性情形中，随机场条件通过强迫核的平方可积性来表达。对于经典的拉普拉斯算子和时空白噪声，端点$\alpha=2$仅在空间维数$d=1$时允许随机场解。本文部分是一篇综述性论文，解释了结果背后的概念和方法。它部分也是一篇研究论文，因为据我们所知，有些结果是新的。
英文摘要：
We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time-space white noise, in the following two cases: (i) With additive fractional time-space white noise (ii) With multiplicative time-space Brownian noise The fractional time derivative is interpreted as the Caputo derivative of order $α\in (0,2]$ and we assume that the Hurst coefficient $H=(H_0,H_1,H_2, ...,H_d)$ of the time-space fractional white noise is in $(\tfrac{1}{2},1)^{1+d}$. We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i). In the multiplicative case (ii) we derive the correct space--time convolution equation and its Wiener chaos expansion. For $1<α\leq2$ a second initial condition is required. The endpoint $α=2$ is treated separately and yields a stochastic wave equation with finite propagation speed. A solution $Y(t,x)$ is called \emph{mild} if $E[Y^2(t,x)] < \infty$ for all $t,x$. In the multiplicative case the random-field condition is expressed by the square integrability of the forcing kernel. For the classical Laplacian and space--time white noise, the endpoint $α=2$ admits a random-field solution only in space dimension $d=1$. This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results are new, to the best of our knowledge.
