$j$ go from $1$ to the dimension $d$ with the summation convention on $i,j$ being enforced. The coefficients $a^{ij}, b^i, c, \sigma^{ijk}, \mu^{ik}, \nu^k$ are functions depending on $(\omega,t,x)$ and the functions $f,f^i, h, g^k$ depend on $(\omega,t,x)$ and the unknown $u$. By considering infinitely many independent Brownian motions $W^k_t$ we cover equations driven by measure-valued processes, for instance, driven by space-time white noise (see Section 3.3). It is worth mentioning that unlike the classical SPDE, we allow the second-order derivatives of the unknown solution $u$ to appear in the stochastic part when $\gamma<1/2$.

As for stochastic differential equations (SDEs), SPDE should be interpreted by its integral form $$\begin{aligned}
&&u(t,x)-u(0,x)\\
&=&\frac{1}{\Gamma(\beta)}\int^t_0
(t-s)^{\beta-1}\left(a^{ij}(s,x)u_{x^ix^j}(s,x)+\cdots+f(s,
u(s,x))\right)ds\\
&&+\frac{1}{\Gamma(1+\beta-\gamma)}
\int^t_0(t-s)^{\beta-\gamma}\left(\sigma^{ijk}(s,x)u_{x^ix^j}(s,x)+\cdots+g^k(s,
u(s,x))\right)dW^k_s.\end{aligned}$$ Similarly one can write down the integral version of SPDE but in the distributional sense with respect to $x$ variable.

We next explain the constraint . A special case of the SPDEs for both and is $$\label{e:1.8}
\partial^\beta_t u (t,x) =\Delta u(t,x) + \partial^\gamma_t \int_0^t g(s,x) dW_s,$$ where $W$ is a one-dimensional Brownian motion. For functions $h_1$ and $h_2$ on $\R_+$, we define its convolution $h_1*h_2$ by $$h_1*h_2 (t)=\int^t_0 h_1(t-s)h_2(s)ds.$$ Let $$k_{\beta}(t):=\Gamma(\beta)^{-1} t^{\beta-1},$$ and define $$I^{\beta} \varphi =k_{\beta}*\varphi := \int_0^t k_\beta (t-s)\varphi (s) ds.$$ 