In ; Stochastic's, SDE, PDE, I have heard the terminology $\textit{"Diffusion Model" or "Diffusion Equation"}$.

The heat equation is sometimes also called the Diffusion Equation ( since it represents the diffusion of heat over some domain, or since its built from Brownian Motion which describes the random trajectories of a brownian particle ). I assume this is just lazy and the heat equation is just a 'very particular diffusion equation'.

Can anyone explain 𝐈𝐧 𝐆𝐞𝐧𝐞𝐫𝐚𝐥 what someone is referring to when they talk about $\textit{'a Diffusion Equation, or the Diffusion Part of the Equation'}$. If anyone can explain this from a mathematical or physical perspective that would be amazing.

Note see Relationship between the diffusion equation and the heat equation for a closely related discussion.

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    $\begingroup$ It's mostly just a naming convention. You get a construction kit for interesting equations in $u_t=cu_x+du_{xx}+f(u)$ where $cu_x$ is a transport term after the transport equation $u_t=cu_x$, $du_{xx}$ the diffusion term for its connection to the Brownian motion and heat equation, as you said, and a local reaction term $f(u)$ for birth-death dynamics, or chemical reaction equations etc. $\endgroup$ Oct 6 '20 at 14:15
  • $\begingroup$ Historic names.. related to its first application... $\endgroup$ Oct 6 '20 at 14:42

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