Kolmogorov forward and backward equations explained

Explain the Kolmogorov forward and backward equations for a diffusion and how they are used.

PDEs for the transition density $p(x,t;y,T)$ of $dX=\mu\,dt+\sigma\,dW$, in the start or end variables. The backward equation acts on the starting point: $p_t+\mu p_x+\tfrac12\sigma^2p_{xx}=0$, the same operator as Black-Scholes, so it is used to price by working back from a payoff. The forward (Fokker-Planck) equation acts on the future point: $p_T=-\partial_y(\mu p)+\tfrac12\partial_{yy}(\sigma^2p)$, and evolves a density forward in time, e.g. for calibration or simulating distributions.