I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which differs from the classical one? To be more specific:
I'm facing the following Fokker-Planck Equation:
$−∂tg(x,t)−Δg(x,t)−div(g(x,t)b(x,t))=0$ with initial condition $g(x,0)=g0(x)$
and where $g_0$ is a density function and $b:ℝn×(0,T)$ is a continuous, bounded and lipschitz in x vector field. I know from the literature, that there is a classical solution $g∈C2,1(ℝn×(0,T))$ of this equation.
What i know aswell is that there is an absolute continuous probability measure (the law of the stochastic process, which satisfies the corresponding SDE), which satisfies this fokker-planck equation in a distributional sense and as a consequence, the density function of this measure satisfies the equation in a weak sense. I now wonder if this density function is a classical solution, because of the statement given above. Does anyone know if this is true? Thanks in advance!