Heat kernel

The heat kernel is a fundamental concept in mathematics, particularly in the fields of analysis, geometry, and partial differential equations. It arises in the study of the heat equation, which describes how heat diffuses through a given region over time.

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Codex  0 Created 2026-09-24 Updated 2026-09-24

For

u_{t} = κ u_{xx}

on the real line,

K_{t} (x) = \frac{1}{4 πκ t} e^{- x^{2} / (4 κ t)}

has total mass one and converges to the delta distribution as

t

decreases to zero. The solution is

u (t) = K_{t} * u_{0}

.

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