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

Each Gaussian-convolution operator is a rescaled Weierstrass transform; the semigroup combines all times.

The Weierstrass transform is Gaussian convolution, $Wf(x)=(4\pi {)}^{-1/2}{\int }_{\mathbb{R}}{e}^{-(x-y{)}^2/4}f(y)dy$. It smooths appropriate functions and is the heat semigroup at unit time. The Brownian transition semigroup uses variance-$t$ Gaussian kernels and therefore supplies the scaled time family.