Let $x = \bar{x} + \epsilon$ where $\bar{x} \sim \mathcal{N}(\mu,\sigma^2)$ and $\epsilon \sim \mathcal{N}(0,\sigma_\epsilon^2(\bar{x}))$ are independent, i.e., the expected value of $x$ is normally distributed, plus an error term normally distributed with a variance depending on the expected value. Then how is $x$ distributed in the following cases:
(1) $\sigma_\epsilon^2(\cdot)\equiv\sigma_\epsilon^2$, then clearly $x\sim \mathcal{N}(\mu, \sigma^2+\sigma_\epsilon^2)$.
(2) $\sigma_\epsilon^2(t)=at$, $a>0$ (you may ignore the case when $t<0$), $x\sim $?
(3) More complicated $\sigma_\epsilon^2(\cdot)$, $x\sim$?