I am trying to think of an intuitive explanation for the mean value property for the heat equation:
$$u(x,t) = \frac{1}{4r^2} \int_{E(x,t,r)} u(y,s) \frac{|y|^2}{|s|^2} dy ds$$
where $E(x,t,r)$ is the heat ball of radius $r $ centered at $(x,t)$.
The analogous formula for the Laplace equation
$$v(x,t) = \frac{1}{|\partial B(x,r)|} \int_{\partial B(x,r)} v(y,s) dy$$
makes sense if you think of Laplacian as a local average (as described here).
My Question. Is there a physical interpretation of the factor $\frac{|y|^2}{|s|^2}$?
I guess one can think of it in terms of "how fast information is propagating in space-time," but that still does not explain why the factor is squared...
I know that there are many questions of a similar title (such as this), but as far as I know, none of them address my question.
