On the conditional distributions of the degenerate diffusion processes

Existence of a density $\pi_{t,s}=\mathbf P\{x_t\in dx\mid y_\tau,\tau\le s\}/dx,\quad s\le t$, for a degenerate multidimensional diffusion process $(x,y)$ as well as the analytical properties of $\pi_{t,s}$ and the filtering equations for $\pi_{t,t}$ are considered.