# Conditions for Gaussianity of SDE

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 $$0\leq s, the follwoing SDE with data has a strong solution $$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?

From the definition of the Itô stochastic integral, 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 one.