The stochastic fractional transport heat equation

Abstract

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.