 Heat kernel

For $u_t=\kappa u_{xx}$ on the real line,has total mass one and converges to the delta distribution as $t$ decreases to zero. The solution is $u(t)=K_t\ast u_0$.

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.