When is the solution to the SDE $dX_t = f(t, X_t)dt + \sigma dW_t$ Gaussian?

Question

I am very new to SDEs, so this question may be basic, but I'm trying to figure it out. Suppose we have and SDE $$dX_t = f(t, X_t)dt + \sigma dW_t$$ where $X_t$ is a 1-dimensional variable, $\sigma$ is a constant, $f(t, X_t)$ is some arbitrary drift function, and $X_0$ follows a standard Gaussian distribution. Does $X_t$ follow standard Gaussian distribution? Why or why not? Does $X_t$ conditioned on $X_0$ follow Gaussian?

Answer 1

If $f(t,X_t)$ is affine, i.e. it is of the form $f(t,X_t) = A(t)X_t + B(t)$, then the solution is always linear.
In the general case I'm not aware of any such result.