Gradient flows and particle representations

Question

I was looking into gradient flows and their particle representations, mostly in the context of probability.

A simple example of this is the continuity equation. Consider evolving a sample $x \sim \rho_0(x)$ using the ODE $\dot x(t) = v(x(t), t)$, with $v$ a vector field. We have $x(0) \sim \rho_0(x)$. The continuity equation tells us that the marginal density evolves in time following the PDE $\frac{\partial{\rho_t(x)}}{\partial{t}} + \nabla \cdot (v(x, t) \rho_t(x))$. This is described for instance here.

In that same reference here, they show two more examples where we have this correspondence between some PDE determining how a density evolves and the particle representation. (One of the examples is the Fokker-Planck equation.)

My question is, is there some intuition to explain this relationship between the particle representations and the PDE for the density? Are there simple rules that determine how can we move from one to the other. In the same video, at some point they say this step requires some creativity, but would still be interested in hearing people's thoughts. Given a nice enough PDE for the time evolution of a density, can we always find the particle representation? And the other way around?

References are appreciated! Just as background, I'm from Computer Science. I'm very comfortable with multivariate Calculus, algebra, etc, but real analysis is not my strong suite (though I did take a class). At this point I'm not that interested in formal proofs / statements as much as getting intuition about what's happening.

Answer 1

You want to derive the Fokker-Planck equation (the drift-diffusion equation for the density) from the Langevin equation (the stochastic differential equation for the position of a particle); this is straightforward if the noise term does not depend on position, see for example these lecture notes.

The "intuition" is that the PDE must conserve the particle density, so it must be of the form of a continuity equation $\partial\rho/\partial t=-\nabla\cdot J(\rho)$. The current density $J$ will for small $\rho$ be linear in $\rho$, and if the density varies slowly it will only contain first derivatives, so $J=J_0 \rho-D\nabla \rho$. The drift term $J_0$ and the diffusion constant $D$ are obtained from the Langevin equation, by calculating the first and second moments of the displacement.

If the noise is position dependent, so the diffusion coefficient $D(x)$ is not constant, an ambiguity appears, do you write $J=J_0\rho-D\nabla \rho$ or do you write $J=J_0\rho-\nabla( D\rho)$ ? This is the Ito-Stratonovich ambiguity, which can only be resolved by a more detailed specification of the random noise, see these notes.