I am trying to prove that the mass is conserved for the Fokker-Planck equation. I know that the Fokker-Planck equation can be written as a continuity equation:
$$ \frac{\partial p}{\partial t}+\nabla \cdot J=0 . $$
where we define $J$ as the vector whose $i$ th component is
$$ J_{i}:=a_{i}(x) p-\frac{1}{2} \sum_{j=1}^{d} \frac{\partial}{\partial x_{j}}\left(b_{i j}(x) p\right) . $$
According to references on the web, if we integrate the FP equation over $\mathbb{R}^{d}$ and integrating by parts on the right hand side of the equation we should get:
$$ \frac{d}{d t} \int_{\mathbb{R}^{d}} p(x, t) d x=0 . $$
implying the conservation of mass. However I don't seem to be able to show that without knowing anything more for $a_i$ and $b_j$. Can someone help me see that? Thanks!