Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq s<T$, the follwoing SDE with data has a strong solution $X_t$: $$ X_t^{x,s} = x + \int_0^t \mu(s,X_s)ds + \sum_{k=1}^m \int_s^t \sigma_k(s,X_s)dW_s^k, $$ where $(W^1,\dots,W^m)$ is an $m$-dimensional Brownian motion. Under what conditions on $\mu$ and the $\sigma_k$ is $X_t^{x,s}$ Gaussian?

Obviously this is true when $\mu$ and the $\sigma_k$ are constants; but how far can we relax our assumptions on these functions?