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Consider the following equation on $[0,T]\times\mathbb{R}^n$ \begin{eqnarray} &\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\ &\rho|_{t=0}=\rho^0, \end{eqnarray} where $V\in C^2(\mathbb{R}^n;\mathbb{R})$. Typical $V$ are of the form $V(x)=x^2$. Further the initial data $\rho^0\in L^1(\mathbb{R}^n;\mathbb{R})$.

Question: How would one prove the existence of solutions of this equation by using standard techniques?

My feeling is that this should work since it is a nice enough parabolic equation, however with somewhat nasty initial data. Such a question has been answered in Jordan-Kinderlehrer-Otto-98 where they use the gradient flow structure of the equation. They assume additionally that the initial data has bounded moments which could be done here as well.

It should also be noted that the above PDE is the Fokker-Planck equation associated to the stochastic differential equation $$\begin{equation} dX_t=-\nabla V(X_t)dt+\sqrt{2} dW_t, \end{equation}$$ where $W_t$ denotes an $n$-dimensional Wiener-process. And since the drift $-\nabla V$ is sufficiently regular, standard results suggest that this equation has a solution. However, I do not yet understand what this result implies for the corresponding Fokker-Planck equation.

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The argument in JKO (short for Jordan-Kinderlehrer-Otto-98) is based on a Lyapunov function for the Fokker-Planck equation. As such, it requires that this Lyapunov function evaluated at $\rho^0$ be finite. (The main result of JKO assumes this condition.) This is a strong constraint on $\rho^0$: it seems to require, e.g., that the support of $\rho^0$ be $\mathbb{R}^n$. For your convenience here is the Lyapunov function from JKO: $$ F(\rho) = \int_{\mathbb{R}^n} \rho \; ( V - \underbrace{(- \log \rho )}_{\text{free energy of $\rho$}} ) dx $$

To answer your question and as far as I can tell: it does not seem possible to prove using standard techniques existence of solutions to the Fokker-Planck equation with what you call "nasty initial data." Recall, in the standard (probabilistic) approach, the Fokker-Planck equation is "well-posed" if:

  • $V$ is of class $C^2$;
  • the transition probability of $X(t)$ admits a probability density function that is twice differentiable for $t>0$; and,
  • the probability law of $X(0)$ has a continuous probability density function on $\mathbb{R}^n$.
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  • $\begingroup$ Nawaf: Thank you very much for the detailed reply. I have a few questions. First of all what do you mean by "U" above? Secondly assuming that one indeed has bounded initial relative entropy (your function F) can we show well-posedness and regularity of solution using standard techniques (semi-group theory for eg) or is the gradient flow structure the only possibility. $\endgroup$ – UPS Mar 22 '15 at 21:55
  • $\begingroup$ I corrected the misprint you mention. $\endgroup$ – Nawaf Bou-Rabee Mar 24 '15 at 19:56
  • $\begingroup$ Concerning your second point, I think their results hold for self-adjoint diffusions, but not beyond that. To read more about self-adjoint diffusions, see my paper with Donev and Vanden-Eijnden: crab.rutgers.edu/~nb361/mypapers/BoDoVa2014.pdf $\endgroup$ – Nawaf Bou-Rabee Mar 24 '15 at 20:08
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The standard references on parabolic PDEs assume bounded coefficients. When dealing with Fokker-Planck equations, however, the coefficients are usually unbounded. The way to get around this is to approximate the coefficients by bounded ones and then derive uniform estimates for the solution and as many moments as you need. There are certainly instances in the literature where this sort of thing has been done. For instance, the case where V is proportional to x^2 arises in molecular models for polymer rheology, and there are quite a few papers on that. You can get started by a Google search with the keyword "Hookean dumbbells."

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