Let $X(t)$ be a wide-sense stationary (WSS) random process with the autocorrelation function $R_{XX} = \sigma^2 e^{-\alpha \tau^2}$. Is $X(t)$ a Gaussian random process?
Here is my try: A random process $Y(t)$ is a Gaussian random process if the samples $Y_1 = Y(t_1), Y_2 = Y(t_2), ..., Y_k=Y(t_k)$ are joint Gaussian random variables for all $k$, and all choices of $t_1, ..., t_k$: $$f_{Y_1, Y_2, ..., Y_k}(y_1, ..., y_k)=\frac{e^{-1/2(\mathbf{x} - \mathbf{m})^T \mathbf{K}^{-1} (\mathbf{x} - \mathbf{m})}}{(2\pi)^{k/2} |\mathbf{K}|^{1/2}}, \tag{1}$$ where $$\mathbf{m} = \begin{bmatrix} m_Y(t_1) \\ \vdots \\ m_Y(t_k) \end{bmatrix}, \mathbf{K} = \begin{bmatrix} C_Y(t_1, t_1) & C_Y(t_1, t_2) & \cdots & C_Y(t_1, t_k) \\ C_Y(t_2, t_1) & C_Y(t_2, t_2) & \cdots & C_Y(t_2, t_k) \\ \vdots & \vdots & & \vdots \\ C_Y(t_k, t_1) & \cdots & & C_Y(t_k, t_k) \end{bmatrix}, \tag{2}$$ where $$C_Y(t_i, t_j) = R_Y(t_i, t_j) - m_Y(t_i) m_Y(t_j), 1 \leq i, j\leq k, \tag{3}$$ and $R_Y(t_i, t_j)$ is the autocorrelation function of $Y(t)$.
Then \begin{align} C_X(t_1, t_2) & = R_X(t_1, t_1) - m_X(t_1) m_X(t_1) \\ & = R_{XX}(0) - [m_X(t_1)]^2 \\ & = \sigma^2 - [m_X(t_1)]^2. \tag{4} \end{align}
Then I don't know how to continue.