What is Infinitesimal generator?

The infinitesimal generator of a time-homogeneous diffusion $dX = \mu(X)\,dt + \sigma(X)\,dW$ is the operator $\mathcal Lf(x) = \lim_{t\downarrow 0} (\E_x[f(X_t)] - f(x))/t = \mu(x)f'(x) + \tfrac12\sigma^2(x)f^{\prime\prime}(x)$ on $C^2$ functions.