# SDE for particle, PDE for the density

Given a particle on the plane we can assume that it follows 2D Brownian motion. On the other hand if there is a lot of such a Brownian particles one can be interested in the evolution of the density of such particles.

How is the PDE for the density derived? I guess it should be similar to the heat equation and there should be connection with a gas dynamics. Could you refer me to the literature?

${d\phi \over dt} = {1\over 2} \nabla^2 \phi$
But this still leaves a scaling invariance, if you rescale space by a certain amount, and time by the same amount squared, you get the same equation back. So the fundamental solution must be proportional to a scaling function: f({x^2\over t}) and you can solve for the solution, which is a spreading gaussian $\phi(x,t) = {1\over (2\pi t)^{d/2}} e^{-{x^2\over 2t}}$, where d is the dimension of space. (a product of one-dimensional Gaussians). This collection of ideas form a circle, because the same distribution is in the definition of Brownian motion, and comes from the central limit theorem.