The name of the operator $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]$ is one-parameter semigroup.

For example one writes $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]f$, where $f$ is a function of $\overrightarrow{v}$.

What is the meaning of this operator?

Does $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]f$ mean $EXP[\frac{ \theta }{2}\frac{\partial^2 f}{\partial \overrightarrow{v}^2}]$ ?