Dirac Delta Function

The Dirac delta function $\delta (x)$ is a generalised function (distribution) defined by two properties:

$$\delta (x)=0\text{for }x\ne 0,{\int }_{-\infty }^{\infty }\delta (x)dx=1$$

It is not a function in the classical sense but acts as a limit of sharply peaked functions (e.g. a Gaussian with vanishing width).

Sifting Property

For any smooth function $f$:

$${\int }_{-\infty }^{\infty }f(x)\delta (x-a)dx=f(a)$$

The delta function “sifts out” the value of $f$ at the point $x=a$.

Use in Diffusion

In the context of this module, $\delta (x)$ appears as the initial condition for the Heat equation: a quantity concentrated at a single point at $t=0$. The solution is then a spreading Gaussian:

$$u(x,t)=\frac{1}{\sqrt{4\pi \alpha t}}\exp \left(-\frac{x^2}{4\alpha t}\right)$$

This is the Green’s function (fundamental solution) of the heat equation. For the Wiener process, the probability density starts as $\delta (w)$ and spreads as a Gaussian with variance $t$ (corresponding to $\alpha =1/2$).

Heat equation | Wiener process | Boundary conditions