$$\label{random-periodic-op}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_3 m_3(t,x)u(t, x) =0\, & x\in  \R^N,\\
 u(t+T, x)=u(t, x+p_j{\bf e_j})=u(t, x),
\end{cases}$$ respectively (see [@ShXi1] and references therein for the relation between nonlocal dispersal operators with Dirichlet type, Neumann type, and periodic boundary conditions, and random dispersal operators with Dirichlet, Neumann, and periodic boundary conditions).

The eigenvalue problems of random dispersal operators with indefinite weight functions have been extensively for more than two decades (see, e.g. [@BeHe; @Boc; @Bo; @BrCoFl; @BrLi; @CaCo1; @Co; @CoCuPo; @He; @He0; @HeKo; @HiKaLa; @KaLoYa; @LoYa; @SeHe] and references therein). For $i=1$ ($i= 2$, or $i= 3$, respectively), recall that a real number $\lambda_1^{r,p}$ ($\lambda_2^{r,p}$ or $\lambda_3^{r,p}$, respectively) is called a [*principal eigenvalue*]{} of (, or , respectively) if (, or , respectively) with $\lambda_1=\lambda_1^{r,p}$ ($\lambda_2=\lambda_2^{r,p}$, or $\lambda_3=\lambda_3^{r,p}$, respectively) has a positive solution (called [*eigenfunction*]{}). The eigenvalue problem , , or are closely related to the following regular eigenvalue problems, $$\label{random-dirichlet-op-eq}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_1m_1(t,x)u(t, x)=\mu_1 u(t, x) \, & x\in  D,\\
 u(t, x)=0, & x\in \p D,\\
 u(t, x)=u(t+T, x),
\end{cases}$$ $$\label{random-neumann-op-eq}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_2 m_2(t,x)u(t, x)=\mu_2 u(t, x) \, & x\in  D,\\
 \frac{\p u}{\p {\bf n}}(t, x)=0, & x\in \p D,\\
 u(t, 