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Nawaf Bou-Rabee
Nawaf Bou-Rabee
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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:

  • $U$$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 $\mathbb{R}^n$.

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:

  • $U$ 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 $\mathbb{R}^n$.

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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Nawaf Bou-Rabee
Nawaf Bou-Rabee
  • 6k
  • 2
  • 20
  • 40

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 I copy down thishere is the Lyapunov function from JKO: $$ F(\rho) = \int_{\mathbb{R}^n} \rho ( U - \underbrace{(- \beta^{-1} \log \rho )}_{\text{free energy of $\rho$}} ) dx $$$$ F(\rho) = \int_{\mathbb{R}^n} \rho \; ( V - \underbrace{(- \log \rho )}_{\text{free energy of $\rho$}} ) dx $$

To answer your questionTo 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:

  • $U$ 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 $\mathbb{R}^n$.

The argument in JKO 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 I copy down this Lyapunov function from JKO: $$ F(\rho) = \int_{\mathbb{R}^n} \rho ( U - \underbrace{(- \beta^{-1} \log \rho )}_{\text{free energy of $\rho$}} ) dx $$

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:

  • $U$ 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 $\mathbb{R}^n$.

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:

  • $U$ 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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Nawaf Bou-Rabee
Nawaf Bou-Rabee
  • 6k
  • 2
  • 20
  • 40

The argument in JKO 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 I copy down this Lyapunov function from JKO: $$ F(\rho) = \int_{\mathbb{R}^n} \rho ( U - \underbrace{(- \beta^{-1} \log \rho )}_{\text{free energy of $\rho$}} ) dx $$

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:

  • $U$ 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 $\mathbb{R}^n$.