The Gaussian Function

Consider, for example, the diffusion equation (2.1) together with the initial condition:

 

and the boundary condition:

 

where we seek a solution in all n-dimensional space. For these boundary conditions to be consistent, we assume the given initial condition satisfy:

 

  

 

   

The property that can be interpreted as a variance suggests that as increases, the Gaussian kernel spreads out and becomes broader and flatter, whereas as decreases towards zero, the Gaussian kernel becomes narrower and sharper, while the total area under the curve remains constant.

 

  

 

Proof

First note that:

 

where . Therefore:

 

and hence:

 

On the other hand:

 

and hence .

  

Proof

First we check the partial differential equation. Lemma 4 together with Lemma 4 implies:

 

Next, property (2.9) of the Gaussian kernel together with Lemma 3 imply:

 

Finally, since both and die out as , it is clear that their convolution dies out as well.

• Discussion
  • Time Evolution of the Solution
  • Probabilistic Interpretation