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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?

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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.

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  • $\begingroup$ Ohh yes thank you! I know that it is the drift of the associated SDE, but can I also call it velocity field? I would like to have a term that is more related to the dynamics of the underlying system and depart from the terms that connect it to the SDE $\endgroup$ Nov 30, 2020 at 23:14
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    $\begingroup$ @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. $\endgroup$ Dec 1, 2020 at 6:26
  • $\begingroup$ Thank you! It seems that this can be called flow velocity vector field according to en.wikipedia.org/wiki/Continuity_equation#Fluid_dynamics So I will name it time-dependent flow velocity vector field. Hope people will accept it :) $\endgroup$ Dec 1, 2020 at 11:43

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