Here I am interested in the heat equation over the domain $\mathbb{R}_+\times\mathbb{R}^d$.
I read this question Green’s Function for the Heat Equation whereby for the heat equation
$$\partial_t u= \Delta u,~~~u(x,0)=u_0$$
we can find the solution at time $t$ by convolving the initial condition $u_0$ with Greens function $G(\cdot,t)$, which solves
$$ \partial_tG=\Delta G,~~~G(\cdot,0)=\delta_0(\cdot), $$
where $\delta_0(x)=0$ if $x\neq 0$ and $1$ if $x=0$.
My question is what is the intuition here ? Are we first solving the PDE for the initial condition of a Dirac mass and then constructing $u_0$ as an infinite sum (or integral) of Dirac masses? If so why is it the Dirac delta function at $0$ which appears not at some other arbitrary element in $\mathbb{R}^d$.
Any other references to the relation between Greens function and the stochastic properties of the underlying particle dynamics associated to the heat equation would be much appreciated too !