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?



Conditions for Gaussianity of SDE
From the definition of the [Itô stochastic integral](https://en.wikipedia.org/wiki/It%C3%B4_calculus#Integration_with_respect_to_Brownian_motion), it is clear that the process $(X\_t)$ will be Gaussian if (i) $\mu(s,\cdot)$ is affine -- that is, $\mu(s,x)=a(s)+b(s)x$ for some regular enough functions $a$ and $b$ and all appropriate $s$ and $x$ and (ii) $\sigma\_k(s,x)$ does not depend on $x$. An example of such a process is the [Ornstein--Uhlenbeck](https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Definition) one.


Other than that, there are probably a few isolated exotic examples. Indeed, more generally, there seem to be few examples of non-affine transformations of Gaussian distributions that are Gaussian as well.


