Dirac delta function
The heat kernel

The heat kernel, defined by

η ε (x) = 1 2 π ε e − x 2 2 ε {\displaystyle \eta _{\varepsilon }(x)={\frac {1}{\sqrt {2\pi \varepsilon }}}\mathrm {e} ^{-{\frac {x^{2}}{2\varepsilon }}}}

represents the temperature in an infinite wire at time t > 0, if a unit of heat energy is stored at the origin of the wire at time t = 0. This semigroup evolves according to the one-dimensional heat equation:

∂ u ∂ t = 1 2 ∂ 2 u ∂ x 2. {\displaystyle {\frac {\partial u}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}u}{\partial x^{2}}}.}

In probability theory, η (x) is a normal distribution of variance ε and mean 0. It represents the probability density at time 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 R, the heat kernel is η ε = 1 (2 π ε) n / 2 e − x ⋅ x 2 ε, {\displaystyle \eta _{\varepsilon }={\frac {1}{(2\pi \varepsilon)^{n/2}}}\mathrm {e} ^{-{\frac {x\cdot x}{2\varepsilon }}},} and has the same physical interpretation, mutatis mutandis. It also represents a nascent delta function in the sense that η → δ in the distribution sense as ε → 0.
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