In themathematical study ofheat conduction anddiffusion, aheat kernel is thefundamental solution to theheat equation on a specified domain with appropriateboundary conditions. It is also one of the main tools in the study of thespectrum of theLaplace operator, and is thus of some auxiliary importance throughoutmathematical physics. The heat kernel represents the evolution oftemperature 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 timet = 0.

The most well-known heat kernel is the heat kernel ofd-dimensionalEuclidean spaceR^d, which has the form of a time-varyingGaussian function,$${\displaystyle K(t,x,y)=\frac{1}{{\left(4\pi t\right)}^{d/2}}\exp \left(-\frac{{\Vert x-y\Vert }^2}{4t}\right),}$$which is defined for all${\displaystyle x,y\in {\mathbb{R}}^d}$ and${\displaystyle t>0}$.^[1] This solves the heat equationfor the unknown function K. Hereδ is aDirac delta distribution, and the limit is taken in the sense ofdistributions, that is, for every functionϕ in the spaceC^∞
_c(R^d) ofsmooth functions with compact support, we have^[2]$${\displaystyle \underset{t\to 0}{lim}{\int }_{{\mathbb{R}}^d}K(t,x,y)\varphi (y)dy=\varphi (x).}$$

On a more general domainΩ inR^d, such an explicit formula is not generally possible. The next simplest cases of a disc or square involve, respectively,Bessel functions andJacobi theta functions. Nevertheless, the heat kernel still exists and issmooth fort > 0 on arbitrary domains and indeed on anyRiemannian manifoldwith boundary, provided the boundary is sufficiently regular. More precisely, in these more general domains, the heat kernel the solution of the initial boundary value problem

To derive a formal expression for the heat kernel on an arbitrary domain, consider the Dirichlet problem in a connected domain (or manifold with boundary)U. Letλ_n be theeigenvalues for the Dirichlet problem of theLaplacian^[3]Letϕ_n denote the associatedeigenfunctions, normalized to be orthonormal inL²(U). The inverse Dirichlet LaplacianΔ⁻¹ is acompact andselfadjoint operator, and so thespectral theorem implies that the eigenvalues ofΔ satisfyThe heat kernel has the following expression:$${\displaystyle K(t,x,y)=\sum _{n=0}^{\mathrm{\infty }}{e}^{-{\lambda }_{n}t}{\varphi }_n(x){\varphi }_n(y).}$$Formally differentiating the series under the sign of the summation shows that this should satisfy the heat equation. However, convergence and regularity of the series are quite delicate.

The heat kernel is also sometimes identified with the associatedintegral transform, defined for compactly supported smoothϕ by$${\displaystyle T\varphi ={\int }_{\mathrm{\Omega }}K(t,x,y)\varphi (y)dy.}$$Thespectral mapping theorem gives a representation ofT in the form thesemigroup^[4][5]

$${\displaystyle T=e^{t\mathrm{\Delta }}.}$$

There are several geometric results on heat kernels on manifolds; say, short-time asymptotics, long-time asymptotics, and upper/lower bounds of Gaussian type.

• Heat kernel signature
• Minakshisundaram–Pleijel zeta function
• Mehler kernel
• Weierstrass transform § Generalizations

• Heat conduction
• Spectral theory
• Parabolic partial differential equations

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