Brownian transition semigroup

For suitable $f$,defines the Brownian transition semigroup. Its generator identity is

The transition density of standard Brownian motion in ${\mathbb{R}}^d$ is the heat kernelThus ${\mathbb{P}}_x(X_t\in A)={\int }_A{p}_t(x,y)dy$.

The area density of ${\mathbb{P}}_z(B_t\in dw,t<T_D)$ for planar Brownian motion killed on its first exit from $D$. Its total mass is the survival probability, not generally one. Integrating this density over time gives the Green function of killed planar Brownian motion.

For a twice differentiable integrable test function $f$ and Brownian motion $W$,is a martingale. This is the one-dimensional generator form of Dynkin's formula.