# Exponential of the Laplacian operator as diffusion equation

Let $u$ be a function on a domain $\Omega$ with some fixed boundary condition.

I have recently seen a notation $e^{\tau \Delta}u$ as meaning the the time evolution of $u$ by diffusion for a time $\tau$. I'm curious where this notation comes from, and more generally, what is meant by a function of a differential operator.

There is an entire branch of mathematics dedicated to the question what is $e^{tA}$ for a differential operator $A$: One parameter Semigroup theory and theory of evolution equations. You could take a look at the Short_Course from Engel and Nagel or directly consult PDE literature (like Chapter 7.4 in Evans' book on PDEs).