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



What is the meaning of the operator $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]$?
This is a symbol which reminds that the solution of linear system of ODEs is represented as an exponential function of a matrix. This symbol represents the solution family of the heat equation, and indeed called a one-parameter semigroup (or semi-dynamical system).


Great introduction in the subject are the books of Engel and Nagel, the [original one](http://www.fa.uni-tuebingen.de/research/publications/1999/one-parameter-semigroups-for-linear-evolution-equations/engel-nagel_one-parameter-semigroups.pdf) and a [short version](http://www.fa.uni-tuebingen.de/research/publications/2006/a-short-course-on-operator-semigroups/A_Short_Course_on_Operator_Semigroups.pdf).


So $ EXP[t\frac{\partial^2}{\partial \overrightarrow{v}^2}]f$ is the solution of the partial differential equation.


$$\partial\_t u(\overrightarrow{v},t)= \frac{\partial^2}{\partial \overrightarrow{v}^2}u(\overrightarrow{v},t)$$
with initial condition 
$$u(\overrightarrow{v},0) = f(\overrightarrow{v})$$
and appropriate boundary conditions depending on the domain and the problem at hand.


