Heat Kernel

The fundamental solution (integral kernel) of the heat (diffusion) equation for a given elliptic operator; it gives the temperature (or density) at a point and time produced by an instantaneous unit source at another point.

The fundamental solution (integral kernel) of the heat equation that propagates initial data forward in time; equivalently the kernel of the heat semigroup exp(tL) for a Laplace-type operator L, providing time-dependent smoothing and short-time geometric asymptotics.