Consider the heat equation $$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^2u}{\partial x^2}$$ with the initial condition $u(0,x)=f(x)$. Any function of the form $E_{x} [f(B_t)]$, where $B_{t}$ represents the stochastic process(the position of a particle undergoing brownian motion) at time t, will satisfy the heat equation subjected to that initial condition. The proof is also there but I am not able to understand the physical intuition behind it. I am considering that heat is in the form of small packets. I understand that the atoms of the material are fixed and they transfer the heat packets to the neighbouring atoms in random directions leading to brownian motion of the heat packets.
I have one explanation(although it isn't complete). Suppose there are 2 blocks of same metal with same mass and volume. The temperature of one block is 30 degrees and another is 60 degrees. After some time, at equilibrium the temperature will be 45 degrees. I think something similar is happening here as well.
$B(t)$ denotes the position of the heat packet starting from position $x$ at time $t$. Now, $f(B_{t})$ denotes the temperature at time 0 of the location where the heat packet from position $x$ reaches at time $t$. There are several locations where the packet can reach at time $t$ since the distribution of the location reached by the packet at time t starting from position x is $N(x,t)$(normal distribution with mean x and variance t). Now, when I take the expected value of $f(B_{t})$, then I am taking the weighted average of the temperatures(recorded at time 0) of the surrounding region(similar to the average taken in case of the metal blocks). The weights or the probability gives a measure of how strong the connection of x is with a particular region in terms of exchange of heat. The more the probability is, the more will be the contribution of that region in altering x's temperature.
Is this explanation correct? Also, why does this averaging thing determine the temperature?