Consider a generic multidimensional diffusion SDE \begin{equation} dX_t = b(X_t,t)dt + \sigma(X_t,t)dW_t. \end{equation} By standard theory $X$ is a Markov process, which defines a Markov semigroup of operators $T_t$ that acts on suitably nice functions $f$
\begin{equation} T_tf(x)= \int f(y) p_t(dy,x) \end{equation}
where $p_t(dy,x)$ are probability measures (ie transition kernels) which satisfy the Chapman-Kolmogorov equation.
When $b$ and $\sigma$ are constant matrices, there is an explicit form for the transition kernels, and thus for the semigroup. This is the Ornstein-Uhlenbeck semigroup. However, I am struggling to find other examples of diffusion SDEs where the analytical form of the transition kernels has been worked out. What are the SDEs with known transition kernels?
This is only a partial answer. Bakry, Gentil and Ledoux in their book Analysis and geometry of partial differential operators give the expression for the transition kernels of the Jacobi and Laguerre diffusion semigroups in the section entitled "Diffusion Semigroups Associated with Orthogonal Polynomials".