I have read in several references that claims the following result: Given a 1-dimensional SDE $$dX_t = a(X_t,t)dt + \sigma(X_t,t) dW_t $$ where $W_t$ is a brownian motion and $\alpha,\sigma:\mathbb R\times \mathbb R\to\mathbb R$ are smooth functions (with $\sigma>0$), then the Fokker-Plank equation $$\frac{\partial }{\partial t}p(x,t) = -\frac{\partial}{\partial x}\left(a(x,t)p(x,t)\right)+\frac{1}{2}\frac{\partial}{\partial x^2}\left(\sigma^2 (x,t) p(x,t)\right),$$ correspond to the equation of the density function of the solution of the SDE (for instance, on the Wikipedia page it is possible to see this result).
I have seen several "derivations" of the Fokker-Plank equation (for instance in the book "Green functions for second-order parabolic integro-differential problems"- M.G. Garroni and J. L. Menaldi). My problem is that in all references that I have seen, they seem to start assuming that the measure $$\mu(A) = \mathbb P(X_t\in A),$$ is absolutely continuous with respect to the Lebesgue measure. Therefore, my question is the following, is it possible for the inverse path? If $p$ satisfies the Fokker-Plank equation is it possible to show that $p$ is the probability density function o the SDE solution $X_t$?
A reference would be enough for me. All the best.