The time-fractional heat equation driven by fractional time-space white noise

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 noise (ii) With multiplicative noise. The fractional time derivative is interpreted as the Caputo derivative of order $\alpha \in (0,2)$ 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 show that there is a unique solution in the Hida space $(\mathcal{S})^{*}$ of stochastic distributions and we show that the solution coincides with the solution of an associated fractional stochastic Volterra equation.Then we give an explicit expression for the solution of this Volterra equation. A solution $Y(t,x)$ is called \emph{mild} if $E[Y^2(t,x)] < \infty$ for all $t,x$. For both the additive noise case and the multiplicative noise case we show that if $\alpha \geq 1$ then the solution is mild if $d=1$ or $d=2$, while if $\alpha < 1$ the solution is not mild for any $d$. The 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 appears to be new.