the heat kernel

Imagine concentrating all your heat into a single point at one instant — an infinitely sharp spike of temperature at the origin — and then watching it spread. What you see is a bump that starts impossibly tall and thin and gradually melts into a wide, low, bell-shaped curve. That spreading bell is the heat kernel: the heat equation's response to a single point source of heat.

On the whole line, the heat kernel is the Gaussian G(x,t) = (1 / sqrt(4 pi k t)) e^(-x^2 / (4 k t)) for t > 0. It is the fundamental solution: it solves u_t = k u_xx with initial data equal to a Dirac delta at the origin (the delta is not an ordinary function but a distribution — an idealized unit spike of zero width). Two features are built in. Its width grows like sqrt(4 k t), the signature diffusive spreading, and its area stays exactly 1 for all time, encoding conservation of total heat. The magic is what it lets you do: to solve the Cauchy problem with any initial temperature f, you convolve f with the kernel, u(x,t) = integral of G(x - y, t) f(y) dy. In words, the temperature at x is a weighted average of all the initial temperatures, with nearby points weighted most and far points weighted by a Gaussian tail.

The heat kernel is one of the most important objects in analysis. Because the Gaussian is infinitely smooth and strictly positive everywhere for any t > 0, the convolution instantly smooths any rough f (the smoothing effect) and instantly feels every point (infinite propagation speed — the Gaussian tail is tiny but never exactly zero). The same object reappears as the transition density of Brownian motion and at the heart of probability. Caveat: the formula above is for the unbounded line; on a finite interval or with boundaries the kernel is different (built from the eigenfunctions or by the method of images), though it plays the same role.