What is Kolmogorov equations, stationary distribution?

For $u(t, x) = \E[g(X_T) \mid X_t = x]$, the Kolmogorov backward equation is $\partial_tu + \mathcal Lu = 0$ with $u(T, \cdot) = g$. The transition density $p(t, y)$ of $X_t$ solves the Kolmogorov forward equation, also called the Fokker–Planck equation, $\partial_tp = \mathcal L^*p = -\partial_y(\mu p) + \tfrac12\partial_{yy}(\sigma^2p)$. A stationary distribution of a Markov process is a law that, taken as the law of $X_0$, is the law of every $X_t$; for a diffusion its density solves $\mathcal L^*p = 0$.