[Paper Review] Uniqueness for a class of stochastic Fokker-Planck and porous media equations

Michael Röckner, Francesco Russo|arXiv (Cornell University)|Sep 1, 2016

Stochastic processes and financial applications10 references4 citations

TL;DR

This paper establishes uniqueness for stochastic Fokker-Planck and porous media equations in the sense of distributions, under minimal assumptions: degenerate and measurable second-order coefficients, multiplicative noise via Brownian motions, and solutions in a large class of progressively measurable random fields. The key result is pathwise uniqueness for both equations using a martingale representation and energy estimates in $H^{-1}$-based norms.

ABSTRACT

The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.

Motivation & Objective

• To establish uniqueness for a class of stochastic Fokker-Planck equations with degenerate and measurable second-order coefficients.
• To extend uniqueness results to stochastic porous media equations in infinite volume, beyond the standard $L^2$-based frameworks.
• To prove uniqueness in the sense of distributions for solutions in a large space of progressively measurable random fields, without requiring integrability in expectation.
• To develop a probabilistic representation framework for stochastic porous media equations using the uniqueness result for the Fokker-Planck equation.
• To generalize existing uniqueness results by relaxing regularity assumptions on coefficients and solution spaces.

Proposed method

• Formulates the stochastic Fokker-Planck equation (1.1) and porous media equation (1.2) in the sense of Schwartz distributions on $\mathbb{R}$.
• Employs Itô-type stochastic multiplication and assumes the noise is driven by independent Brownian motions $W^i_t$ and a deterministic drift $e^0(\xi)t$.
• Uses $H^{-1}$-multiplier theory to handle degenerate and measurable coefficients $e^i$, ensuring well-definedness of stochastic terms.
• Applies a mollification procedure via $\phi_\varepsilon$ to approximate solutions and derive energy estimates in $H^{-1}$-norm.
• Derives a martingale representation for the $H^{-1}$-norm of the difference of two solutions and applies Itô's formula to the squared norm.
• Uses the Lipschitz continuity of $\psi$ and a Young-type inequality to control nonlinear terms, leading to a Gronwall-type inequality.

Experimental results

Research questions

• RQ1 Can uniqueness be established for stochastic Fokker-Planck equations when the second-order coefficients are merely measurable and possibly degenerate?
• RQ2 Is pathwise uniqueness possible for stochastic porous media equations in infinite volume without requiring solution integrability in expectation?
• RQ3 Can the solution class be significantly enlarged beyond $L^2$ or $L^1 \cap L^\infty$ while preserving uniqueness in the sense of distributions?
• RQ4 Does the use of $H^{-1}$-norm energy estimates and martingale representation allow for uniqueness under minimal regularity assumptions on coefficients?
• RQ5 Can the uniqueness result for the Fokker-Planck equation be leveraged to prove uniqueness for the related porous media equation?

Key findings

• Uniqueness holds for the stochastic Fokker-Planck equation (1.1) under the assumption that the second-order coefficients $a$ are bounded and progressively measurable, and the noise coefficients $e^i$ are $H^{-1}$-multipliers.
• The solution class for the Fokker-Planck equation includes all progressively measurable random fields $z$ such that $\int_{[0,T]\times\mathbb{R}} z^2(s,\xi) \, ds \, d\xi < \infty$ a.s.
• For the porous media equation (1.2), uniqueness is proven in the class of progressively measurable random fields $X$ satisfying $\int_{[0,T]\times\mathbb{R}} X^2(s,\xi) \, ds \, d\xi < \infty$ a.s., without requiring integrability in expectation.
• The proof relies on an energy estimate in the $H^{-1}$-norm of the difference of two solutions, leading to a Gronwall-type inequality via the martingale term $M_t$.
• The key estimate shows $\|X(t,\cdot)\|_{H^{-1}}^2 \leq M_t + C \int_0^t \|X(s,\cdot)\|_{H^{-1}}^2 \, ds$ a.s., where $C$ depends on the multiplier norms of $e^i$ and $e^0$.
• The result is robust to degeneracy: even if $a$ is degenerate or $e^i$ are only measurable, uniqueness holds as long as they are $H^{-1}$-multipliers.