# Fokker--Planck equation - naming a vector field

A Fokker Planck equation for the prob. density $$\rho$$ may be written in the form of a continuity equation $$\frac{\partial \rho(x,t)}{\partial t} = - \nabla \cdot \left[ g(x,t) \rho(x,t) \right].$$

The term $$\left[ g(x,t) \rho(x,t) \right]$$ is often called the probability current or the probability flux.

I was wondering whether there is a name for the term $$g(x,t) .$$ It is a vector field, but is there some more specific or descriptive characterisation for this term?

It is called the drift term (here denoted as $$\mu(X_t, t)$$ https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation. You have written the form with zero diffusion term, more generally: $$\frac{\partial\rho(x,t)}{\partial t} = -\nabla [g(x, t) \rho(x,t)] + \nabla [D(x,t) \nabla p(x,t)]$$ Where $$D = D_{ij}$$ is the diffusion tensor.
• @ii.iiii, i wouldn't say so in general, it the case of constant in space and time field, this does really resemble the advection equation : $$\partial_t \rho = -g \nabla \rho$$, and $g$ would be a speed of wave, but in general, there is a term $$(\nabla g) \rho$$, which would make it not like the wave equation with coordinate and time-dependent velocity. Dec 1, 2020 at 6:26