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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 !

Bernard
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bus busman
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  • This has helped somewhat https://math.stackexchange.com/questions/89785/what-does-the-heat-kernel-in-the-heat-equation-represent-ux-t?rq=1 – bus busman Nov 25 '21 at 20:17
  • Not quite, the Green's function solves $$(\partial_t-\Delta)G = \delta_0$$ This distinction is important because of what a Green's function does - $G$ acts as the inverse operator (matrix) to the heat operator and $\delta_0$ is the continuous version of the identity matrix. This relates to the fact that convolution is really an operation that describes a choice of dirac-masses-at-each-point basis and the linear combination of a function in that continuous basis. – Ninad Munshi Nov 25 '21 at 23:37
  • @NinadMunshi can you expand a bit on what you mean/how the linked question is wrong, and maybe suggest some reading on the topic - more for ideas than specifics I guess.. – bus busman Nov 26 '21 at 09:16
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    @busbusman : what you describe in your question as $G$ is the heat kernel also often called the *fundamental solution*. The intuition behind that is exactly with you think. – Kurt G. Nov 26 '21 at 19:52
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    You basically write the solution as a "continuous linear combination" of the solutions you would get with Dirac masses at different points. The fundamental solution itself has the Dirac mass at the origin but the "combination" process stitches together the results of putting Dirac masses elsewhere. – Ian Nov 27 '21 at 02:24

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