Editing Dirac delta function (section)

=====The heat kernel=====
The [[heat kernel]], defined by

<math display="block">\eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}}</math>

represents the temperature in an infinite wire at time {{math|1=''t'' > 0}}, if a unit of heat energy is stored at the origin of the wire at time {{math|1=''t'' = 0}}. This semigroup evolves according to the one-dimensional [[heat equation]]:

<math display="block">\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}.</math>

In [[probability theory]], {{math|1=''η<sub>ε</sub>''(''x'')}} is a [[normal distribution]] of [[variance]] {{mvar|ε}} and mean {{math|0}}. It represents the [[probability density function|probability density]] at time {{math|1=''t'' = ''ε''}} of the position of a particle starting at the origin following a standard [[Brownian motion]]. In this context, the semigroup condition is then an expression of the [[Markov property]] of Brownian motion.

In higher-dimensional Euclidean space {{math|'''R'''<sup>''n''</sup>}}, the heat kernel is
<math display="block">\eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}},</math>
and has the same physical interpretation, {{lang|la|[[mutatis mutandis]]}}. It also represents a nascent delta function in the sense that {{math|''η<sub>ε</sub>'' → ''δ''}} in the distribution sense as {{math|''ε'' → 0}}.