 Heat kernel

For $D>0$ and $t>0$, the Euclidean heat kernel for $u_t=D\Delta u$ on ${\mathbb{R}}^d$ isIt is the fundamental solution from a unit Dirac delta distribution. Its integral is one, its coordinate variances are $2Dt$, and $K_t\ast K_s=K_{t+s}$. For suitable initial data, $u(\cdot ,t)=K_t\ast u_0$ solves the heat equation. The approximate identity property recovers the initial data in Lp norm for $1\le p<\infty$, and locally uniformly for bounded continuous data.^[ref]

The formula quantifies spreading: the width grows like $\sqrt{Dt}$ while the peak falls like $t^{-d/2}$. Its strict positivity yields infinite propagation speed for nonzero nonnegative initial data. On an interval, boundary conditions require a different kernel, such as the Dirichlet heat kernel on an interval or Neumann heat kernel on an interval; their mass behavior records whether the boundary absorbs or reflects. The Gaussian heat kernel for standard Brownian motion uses $D=1/2$, which explains its different factor of two.

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.