Talk:Heat kernel

Puzzling expression in introduction

The Introduction begins as follows:

"In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time t = 0.

Fundamental solution of the one-dimensional heat equation. Red: time course of $\Phi (x,t)$. Blue: time courses of $\Phi (x_{0},t)$ for two selected points. Interactive version.

"The most well-known heat kernel is the heat kernel of d-dimensional Euclidean space R^d, which has the form of a time-varying Gaussian function,

$K(t,x,y)=\exp \left(t\Delta \right)(x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-\|x-y\|^{2}/4t}\qquad (x,y\in \mathbb {R} ^{d},t>0)\,$"

What does exp(∆t) (x,y) mean here?

As a mathematician, my understanding is that since ∆ is an operator, then so is exp(∆t) also an operator.

So, what does it mean to apply an operator on a function space to the ordered pair (x,y) of points in R^d ???