Time derivative of the mapping $t \mapsto P_tf(x)=\mathbb{E}^x(f(X_t))$ - infinitesimal generator

Question

Can anybody please explain equation $1$ in this answer in a simpler form? Also i cannot understand how from equation $1$ we can see that $u$ is the solution to the heat equation.

Answer 1

Hint

\begin{align} \frac{\mathrm d }{\mathrm dt}P_tf(x)&=\lim_{h\to 0}\frac{P_{t+h}f(x)-P_tf(x)}{h}\\ &=\lim_{h\to 0}P_t\left(\frac{P_hf(x)-f(x)}{h}\right)\\ &=P_t\left(\lim_{h\to 0}\frac{P_hf(x)-f(x)}{h}\right)\\ &=P_tAf(x)\\ &=AP_tf(x). \end{align} I let you justify each equality as a homework. For your other question, one can prove that the infinitesimal generator of the Brownian motion if given by $$Af(x)=\frac{1}{2}\Delta f(x).$$ Do it as a homework if this is not clear for you.