I can see how the heat equation (say on $\mathbb{R}^d$) : $\partial_t \rho=\Delta\rho$, can be viewed as a $L_2$ gradient flow of the energy
$$ E(\rho)=\frac{1}{2}\int_{\mathbb{R}^d} |\nabla \rho(x)|^2dx. $$
What is the $L_2$ gradient flow of the Fokker-Planck (in fact Langevin) equation :
$$ \partial_t \rho=\text{div}(\rho\nabla f)+\Delta\rho, $$
for some nice enough function $f:\mathbb{R}^d\to \mathbb{R}$.