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
$$\int_0^\infty f(x)\big(p(t,x)-\rho(x)\big)dx = \int_0^t\int_0^\infty\left(b(s,x)f'(x)+\frac{\sigma(s,x)^2}{2}f''(x)\right)p(s,x)dxds,\quad \forall t\ge 0,~ f\in C_b^2(\mathbb R_+)\quad \quad (\ast).$$
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?
$$\partial_t p(t,x) = \frac{1}{2}\partial_{xx}^2\big(\sigma^2(t,x)p(t,x)\big)- \partial_x\big(b(t,x)p(t,x)\big),\quad \forall (t, x)\in(0,\infty)\times(0,\infty),$$
$$p(0,\cdot)=\rho,\quad p(\cdot,0)=0.$$
PS : Thanks for the comments below. Let me describe quickly the motivation. Consider the stochastic differential equation: $X_0\sim \rho(x)dx$ and
$$dX_t = {\bf 1}_{\{X_t>0\}}\big(b(t,X_t)dt + \sigma(t,X_t)dW_t\big),\quad \forall t>0.$$
We may show rigorously that $X_t{\bf 1}_{\{X_t>0\}}$ admits a sub-probability density $p(t,\cdot)$. Further, applying Ito's formula, we find $(\ast)$ is satisfied by for test functin $f\in C_b^2(\mathbb R_+)$. If $p$ is regular enough, then $p$ solves the F-P equation. Now, knowing the F-P equation admits a unique classical solution, can we say the uniqueness of $p$ satisfying $(ast)$? In other words, does the uniqueness of the classical solution implies that of the "weak" solution? If so, then we can also derive the uniqueness of $X$.
