It is wellknown that the linear FokkerPlanck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\right) \label{1}\tag{1}$$ where $\rho_\infty$ (say for instance of the form $\rho_\infty(x) \propto \mathrm{e}^{V(x)}$ for some smooth and convex potential $V(x)$ growing sufficiently fast at infinity) is the unique equilibrium distribution to which the solution of (\ref{1}) converges, admits a family of Lyapunov functionals of the form (where $\phi$ is some convex function fulfilling certain properties) $$\mathrm{H}_\phi[\rho] = \int_{\mathbb R} \phi\left(\frac{\rho}{\rho_\infty}\right) \rho_\infty \,\mathrm{d} x \label{2}\tag{2}$$ for the study of the convergence to equilibrium problem, see for instance this monograph. However, I am wondering if there are references for investigation/study of the largetime convergence behavior of the following nonlinear FokkerPlanck type equation $$\partial_t \rho = \partial_x \left(\mathcal{F}[\rho]\, \partial_x\left(\frac{\rho}{\mathcal{F}[\rho]}\right)\right) \label{3}\tag{3}$$ where $\mathcal{F} \colon \rho \in \mathcal{P}(\mathbb R) \to \mathcal{F}[\rho] \in \mathcal{P}(\mathbb R)$ is a sort of "quasistationary distribution" for the PDE (\ref{3}) with $\mathcal{F}[\rho_\infty] = \rho_\infty$. Here quasistationarity rough means (loosely speaking) that $\mathcal{F}[\rho]$ "takes the same form" as the true equilibrium $\rho_\infty$ (for example, they are both Gaussian but with different variance or they are both exponential distributions with different mean values). I am wondering if there some recent or classical reference for the investigation of the large time behavior of such type of nonlinear FokkerPlanck equation, especially the construction of Lyapunov functionals. I have to admit that my question is a not very clear as the analysis of (\ref{3}) will depend on the specific choice of the map/nonlinearity $\mathcal{F}[\cdot]$, but I am hoping that some references pointing to analysis of FokkerPlanck type equations with the very specific structure as indicated in (\ref{3}) can be found.
1 Answer
Let me rewrite the equation (3) as $$ \partial_t \rho = \partial_x \left(\rho\, \partial_x\log\left(\frac{\rho}{\mathcal{F}[\rho]}\right)\right) \label{3'}\tag{3'}. $$ Then, this equation is a gradient flow with respect to the metric tensor after Otto inducing the Wasserstein distance iff there exists a driving free energy $\mathcal{E} : \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}$ such that its variational derivative satisfies $$ \mathcal{E}'(\rho) = \log\left(\frac{\rho}{\mathcal{F}[\rho]}\right) + C, $$ where the constant $C$ does not matter and could even depend on $\rho$.
In the first case with $\mathcal{F}[\rho]\equiv \rho_\infty$, one gets $$ \mathcal{E}(\rho) = H_{\phi}[\rho] \quad\text{with}\quad \phi(r) = r \log r r + 1 . $$ A more interesting case is for some potential energy $V:\mathbb{R}^d \to \mathbb{R}$ and symmetric interaction energy $W:\mathbb{R^d}\times\mathbb{R^d} \to \mathbb{R}$ the map $$ \mathcal{F}[\rho](x) = Z[\rho]^{1} \exp\left( V(x) \int W(x,y) \rho(y) dy \right) \quad\text{with}\quad Z[\rho] := \int \exp\left( V(x) \int W(x,y) \rho(y) dy \right) dx .$$ Then, upto a $\rho$dependent constant, the free energy is given by $$ \mathcal{E}(\rho) = \int \rho \log \rho\, dx + \int V(x) \rho(x) \, dx + \frac{1}{2} \int \int W(x,y) \rho(x)\rho(y)\, dx\, dy . $$ This is the classic McKeanVlasov model and the free energy above consisting of entropy, potential energy and interaction energy is studied a lot in different areas (gradient flows, density functional theory, statistical mechanics).
Depending on your map $\mathcal{F}$ there might be other free energies. The above one is the most common in the literature of meanfield limits for interacting particle systems, where it is usually even assumed that $W(x,y)=w(xy)$ for some even $w:\mathbb{R}^d \to \mathbb{R}$.

$\begingroup$ Hello Andre, first thanks for your answer and post. However, the map $\mathcal{F}$ I have in mind is not the classical one that you wrote (and was wellknown to be honest). I am wondering if there exists a general guidance for the construction of the Lyapunovtype functional (for general nonlinearity $\mathcal{F}[\cdot]$), or one has to deal with the specific $\mathcal{F}$ "case by case". What I have in my project is actually a discrete version of equation (3) where the state space is $\mathbb N$... $\endgroup$– Fei CaoJul 31, 2023 at 22:10

$\begingroup$ In this generality, the best you can try is to find a free energy functional that satisfies the unnumbered equation above $\mathcal{E}'(\rho) = \dots$. Something similar is possible for discrete models. Together with Joe Conlon, I once studied a class of models on $[0,\infty)$ and also $\mathbb{N}$ that have similar properties to McKeanVlasov: arXiv:1711.00782, see in particular Section 6 about the gradient flow structures and Remark 1.4 about its relation to McKeanVlasov. Otherwise, I don't know of any general theory beyond what I wrote above. $\endgroup$ Aug 1, 2023 at 20:32

$\begingroup$ Thank you very much for your reply. I will wait to see if new answers will come out before the end of the bounty award deadline $\endgroup$– Fei CaoAug 2, 2023 at 2:26