TITLE: What physical information does the mean value property of heat equation convey?
QUESTION [9 upvotes]: I'm reading through the Evans' book on PDE, the chapter on heat equation.
The definitions are the same as here.
I see that mean value property of heat equation is useful for proving maximum principle and various uniqueness results, but I'm curious are there any physical interpretations of definition of heat ball and mean value property itself?
It seems to me that definition of heat ball $E(\mathbf x, t; r)$ should have something to do with an idea of bringing a point source to the point $\mathbf x$
at the time $t$, but the fact that we are looking only at the past (points $(y, s)$ of the space-time with $s \leq t$) confuses me.
Can someone help?

REPLY [3 votes]: Mean Value Property
The physical process behind heat transfer is Brownian motion of particles. Brownian motion is a type of martingale. Martingales are random variables that satisfy the mean value property (as such, they model harmonic functions.)
Intuitively speaking, heat is the random motion of particles. These particles have no reason to pick a favorite direction, so therefore as an aggregate they travel in a harmonic fashion.
Heat Ball
Unlike the name suggests, I don't think of a heat ball as a spatial object. It's more like the information available to a particle, based on the particles it could have interacted with in the past.