 Solution

Let $u(t,x)=P_{1-t}f(x)$ for $t<1$. Differentiating the Brownian transition semigroup in the spatial variables is justified by the bounded gradient and givesThe heat equation for this semigroup is the backward equation ${\partial }_{t}u+\frac{1}{2}\Delta u=0$. Applying the multidimensional Itô formula yieldsThe finite-variation term vanishes. ThusInitially this calculation holds on $[0,1-\epsilon ]$. The vector of stochastic integral coefficients has norm at most $K$, so the Itô isometry lets the integrals extend to time one. Also $P_{1-t}f(X_t)\to f(X_1)$ in $L^2$ by the Lipschitz function bound and continuity of Brownian motion. The same representation therefore holds on the entire interval.