Let $(Y_t)$ be a stochastic process solution to the SDE $$dY_t= \lambda(Y_t,t) dt + \sigma(Y_t,t) dB_t. $$
If we know that $(Y_t)$ is a Gaussian process, what does it inform us on the drift $\lambda$ and the volatility $\sigma$ ? To my knowledge, the only link between the distribution of $(Y_t)$ and its drift/volatility is Fokker-Planck equation: if $p(x,t)$ is the density function of $(Y_t)$, we have
$$\frac{\partial}{\partial t} p(x, t)=-\frac{\partial}{\partial x}[\lambda(x, t) p(x, t)]+\frac 1 2 \frac{\partial^{2}}{\partial x^{2}}[\sigma^{2}\left(x, t\right)p(x, t)]$$
I tried "brut-forcing" by plugging in a Gaussian density in the PDE and try to see what contraints on the drift and volatility we get, without much success.
EDIT: you can assume $(Y_t)$ to be one-dimensional for simplicity, but a general answer would be more than welcome.