Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying

$$\int_0^\infty f(x)\big(p(t,x)-\rho(x)\big)dx = \int_0^t\int_0^\infty\left(b(s,x)f'(x)+\frac{\sigma(s,x)^2}{2}f''(x)\right)p(s,x)dxds,\quad \forall t\ge 0,~ f\in C_b^2(\mathbb R_+).$$

Assuming $\rho\ge 0$ ($\int_0^\infty \rho dx=1$), $b\ge 0$, $\sigma>0$ are nice enough (bounded, Lipschitz, elliptic), what is the "minimal condition" that ensures the uniqueness of $p$? In addition, what is the "minimal condition" under which $p$ is the classical solution of the Fokker-Plank equation below?

$$\partial_t p(t,x) = \frac{1}{2}\partial_{xx}^2\big(\sigma^2(t,x)p(t,x)\big)- \partial_x\big(b(t,x)p(t,x)\big),\quad \forall (t, x)\in(0,\infty)\times(0,\infty),$$

$$p(0,\cdot)=\rho,\quad p(\cdot,0)=0.$$