In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time t = 0.
Definition
The most well-known heat kernel is the heat kernel of d-dimensional Euclidean space Rd, which has the form of a time-varying Gaussian function,
K ( t , x , y ) = 1 ( 4 π t ) d / 2 exp ( − ‖ x − y ‖ 2 4 t ) , {\displaystyle K(t,x,y)={\frac {1}{\left(4\pi t\right)^{d/2}}}\exp \left(-{\frac {\left\|x-y\right\|^{2}}{4t}}\right),}
which is defined for all x , y ∈ R d {\displaystyle x,y\in \mathbb {R} ^{d}} and t > 0 {\displaystyle t>0} . This solves the heat equation
{ ∂ K ∂ t ( t , x , y ) = Δ x K ( t , x , y ) lim t → 0 K ( t , x , y ) = δ ( x − y ) = δ x ( y ) {\displaystyle \left\{{\begin{aligned}&{\frac {\partial K}{\partial t}}(t,x,y)=\Delta _{x}K(t,x,y)\\&\lim _{t\to 0}K(t,x,y)=\delta (x-y)=\delta _{x}(y)\end{aligned}}\right.}
for the unknown function K. Here δ is a Dirac delta distribution, and the limit is taken in the sense of distributions, that is, for every function ϕ in the space C∞c(Rd) of smooth functions with compact support, we have
lim t → 0 ∫ R d K ( t , x , y ) ϕ ( y ) d y = ϕ ( x ) . {\displaystyle \lim _{t\to 0}\int _{\mathbb {R} ^{d}}K(t,x,y)\phi (y)\,dy=\phi (x).}
On a more general domain Ω in Rd, such an explicit formula is not generally possible. The next simplest cases of a disc or square involve, respectively, Bessel functions and Jacobi theta functions. Nevertheless, the heat kernel still exists and is smooth for t > 0 on arbitrary domains and indeed on any Riemannian manifold with boundary, provided the boundary is sufficiently regular. More precisely, in these more general domains, the heat kernel is the solution of the initial boundary value problem
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