by $$\label{e:1.1}
\frac{\partial^\beta f(t)}{\partial t^\beta}=\frac{1}{\Gamma (1-\beta)} \frac{d}{dt}
\int_0^t (t-s)^{-\beta}  \left(f(s)-f(0)\right) ds$$ where $\Gamma$ is the Gamma function defined by $\Gamma(\lambda):=
\int^{\infty}_0 t^{\lambda-1} e^{-t}dt$.

Fractional diffusion equations are becoming popular in many areas of application [@GM; @KS; @LS; @MK; @P; @SBMW; @V]. So far, on the basis of either deterministic or probabilistic methods, the study of fractional calculus is mainly restricted to deterministic equations; see [@EIK; @MS; @P; @Za] and the references therein. In this paper, we introduce and investigate a class of stochastic partial differential equations (SPDEs) with fractional time derivatives.

The SPDEs with fractional time derivative that we are going to study in this paper naturally arise from the consideration of the heat equation in a material of thermal memory. Let $u(t,x), e(t,x)$ and $\vec F(t,x)$ denote the body temperature, internal energy and flux density, respectively. Then the relations $$\label{heat 1}
\frac{\partial e}{\partial t}(t,x)=-{\rm div} \, \vec  F,$$ $$e(t,x)=\beta u(t,x), \quad \vec F(t,x)=-\lambda \nabla u(t,x) , \quad\quad
\beta, \lambda >0$$ yield the classical heat equation $\beta \frac{\partial u}{\partial t}=\lambda \Delta u$. According to the law of the classical heat equation, the speed of the heat flow is infinite. However in real modeling, the propagation speed can be finite because 