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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.

An integral transform constructs a new function by integrating an input against a kernel:The kernel $K$ specifies how each input location contributes to an output indexed by $s$. Whenever the integrals exist, linearity gives $T(af+bg)=aTf+bTg$. The function spaces, convergence conditions and normalization are part of the definition; a formal kernel alone need not define an invertible or even everywhere-defined linear operator.^[ref]

The Laplace transform uses $K(s,t)=e^{-st}$ on nonnegative time. For $f(t)=e^{-at}$ it gives $1/(s+a)$ when $\mathrm{Re}(s+a)>0$. The Fourier transform decomposes a function into oscillatory frequencies, while the Mellin transform is adapted to changes of scale. A smoothing Gaussian kernel instead gives the Weierstrass transform.

The useful operation is often a change of representation: the Laplace transform of a derivative turns a constant-coefficient differential equation into an algebraic equation with initial data. The inverse step then requires a theorem, such as the Bromwich inversion formula, with its own regularity and convergence assumptions.

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.