One can define a correlation by making convolutions of two radially symmetric functions. \begin{equation*} C_0(r)=B_1*B_2(r) [B_1*B_2(0)]^{-1} \end{equation*} On the other hand, one can also construct a correlation based on a generalised diffusion equation. It is widely adapted in background matrix modelling in data assimilation. Give here an 1D example of equation: \begin{equation*} \frac{\partial \eta}{\partial t}-\kappa \frac{\partial^2 \eta}{\partial z^2}=0 \quad \textrm{with initial condition} \quad \eta(z,0) \end{equation*}
The solution of this equation \begin{equation*} \eta(z,T)=\frac{1}{\sqrt{4 \pi \kappa T}} \int_{z’} e^{-(z-z’)^2/4\kappa T} \eta(z’,0)dz’ \end{equation*} Define also a correlation on the location $ z$ (T is seen as a parameter).Furthermore, this solution could also be obtained by a convolution product.
Is there an equivalence between these two definitions. For example, can all the solutions obtained from a diffusion equation be expressed under a form of convolution?
