Wigner semicircle distribution This is continuous and compactly supported, but not a mollifier because it is not smooth. Semigroups Nascent delta functions often arise as convolution semigroups. This amounts to the further constraint that the convolution of ηε with ηδ must satisfy for all ε, . Convolution semigroups in L1 that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction. In practice, semigroups approximating the delta function arise as fundamental solutions or Green's functions to physically motivated elliptic or parabolic partial differential equations. In the context of applied mathematics, semigroups arise as the output of a linear time-invariant system. Abstractly, if A is a linear operator acting on functions of x, then a convolution semigroup arises by solving the initial value problem in which the limit is as usual understood in the weak sense. Setting gives the associated nascent delta function. Some examples of physically important convolution semigroups arising from such a fundamental solution include the following. The heat kernel The heat kernel, defined by represents the temperature in an infinite wire at time t > 0, if a unit of heat energy is stored at the origin of the wire at time t = 0. This semigroup evolves according to the one-dimensional heat equation: In probability theory, ηε(x) is a normal distribution of variance ε and mean 0. It represents the probability density at time of the position of a particle starting at the origin following a standard Brownian motion. In this context, the semigroup condition is then an expression of the Markov property of Brownian motion. In higher-dimensional Euclidean space Rn, the heat kernel is and has the same physical interpretation, mutatis mutandis. It also represents a nascent delta function in the sense that in the distribution sense as . The Poisson kernel The Poisson kernel is the fundamental solution of the Laplace equation in the upper half-plane. It represents the electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the Cauchy distribution and Epanechnikov and Gaussian kernel functions. This semigroup evolves according to the equation where the operator is rigorously defined as the Fourier multiplier Oscillatory integrals In areas of physics such as wave propagation and wave mechanics, the equations involved are hyperbolic