What is Markovian projection?

The Markovian projection of an Itô process $dS_t=\mu_tS_t\,dt+ \sigma_tS_t\,dW_t$, with $\sigma_t$ random, is the local volatility model with $\sigma_{\mathrm{loc}}^2(t,K)=\E\bigl[\sigma_t^2\mid S_t=K\bigr]$; by Gyöngy’s theorem, $S_t$ has the same law in both at every date.