Under suitable assumptions on $\beta:\mathbb{ R}\!\to\!\mathbb{ R}, \,D:\mathbb{ R}^d\!\to\!\mathbb{ R}^d$ and $b:\mathbb{ R}^d\!\to\!\mathbb{ R}$, the nonlinear Fokker-Planck equation $u_t-\Delta\beta(u)+{ \rm div}(Db(u)u)=0$, in $(0,\infty)\times\mathbb{ R}^d$ where $D=-\nabla\Phi$, can be identified as a smooth gradient flow $\frac{ d^+}{ dt}\,u(t)+\nabla E_{ u(t)}=0$, $\forall t>0$. Here, $E:\mathcal{ P}^*\cap L^\infty(\mathbb{ R}^d)\to\mathbb{ R}$ is the energy function associated to the equation, where $\mathcal{ P}^*$ is a certain convex subset of the space of probability densities. $\mathcal{ P}^*$ is invariant under the flow and $\nabla E_u$ is the gradient of $E$, that is, the tangent vector field to $\mathcal{ P}$ at $u$ defined by $\left<\nabla E_u,z_u\right>_u={ \rm diff}\,E_u\cdot z_u$ for all vector fields $z_u$ on $\mathcal{ P}^*$, where $\left<\cdot,\cdot\right>_u$ is a scalar product on a suitable tangent space $\mathcal{ T}_u(\mathcal{ P}^*)\subset\mathcal{ D}'(\mathbb{ R}^d)$.
Categories: math.PR math.AP