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Quantitative sensitivity analysis for Fokker-Planck equation with respect to the Wasserstein distance

Martin Morange

TL;DR

The paper addresses the problem of quantifying how solutions to the linear Fokker-Planck equation depend on a parameter $a$ in the Wasserstein metric $\mathcal{W}_p$ for $p \ge 2$. It establishes a quantitative upper bound on $\mathcal{W}_p^p(\rho(t,\cdot,a'),\rho(t,\cdot,a))$ that involves the initial distance and a term proportional to $|a'-a|^p$, with explicit growth rate constants. Two independent proofs are provided: one via differentiation of the Kantorovich dual formulation, and another via a synchronous coupling of the associated SDEs, both leading to the same Grönwall-type bound with explicit constants. The results are specialized to the overdamped Langevin process, yielding a bound that also captures convergence to the invariant measure and parameter-induced discrepancies, including a variant with thermodynamic parameter $\beta$. Overall, the work provides quantitative stability of PDE flows under parameter perturbations in Wasserstein space, with potential applications in uncertainty quantification and sampling methods.

Abstract

We analyze the sensitivity of solutions to the Fokker-Planck equation with respect to some unknown parameter. Our main result is to provide quantitative upper bounds for the $p$-Wasserstein distance $\mathcal{W}_p$ between two solutions with different parameters, for every $p \geq 2$. We are able to give two proofs of this result, the first relying on synchronous coupling between two solutions of an SDE, and another one that relies on the differentiation of Kantorovitch dual formulation of optimal transport. We also provide more specific bounds in the case of the overdamped Langevin process, for which we are able to compare convergence to the invariant measure and sensitivity to the parameter.

Quantitative sensitivity analysis for Fokker-Planck equation with respect to the Wasserstein distance

TL;DR

The paper addresses the problem of quantifying how solutions to the linear Fokker-Planck equation depend on a parameter in the Wasserstein metric for . It establishes a quantitative upper bound on that involves the initial distance and a term proportional to , with explicit growth rate constants. Two independent proofs are provided: one via differentiation of the Kantorovich dual formulation, and another via a synchronous coupling of the associated SDEs, both leading to the same Grönwall-type bound with explicit constants. The results are specialized to the overdamped Langevin process, yielding a bound that also captures convergence to the invariant measure and parameter-induced discrepancies, including a variant with thermodynamic parameter . Overall, the work provides quantitative stability of PDE flows under parameter perturbations in Wasserstein space, with potential applications in uncertainty quantification and sampling methods.

Abstract

We analyze the sensitivity of solutions to the Fokker-Planck equation with respect to some unknown parameter. Our main result is to provide quantitative upper bounds for the -Wasserstein distance between two solutions with different parameters, for every . We are able to give two proofs of this result, the first relying on synchronous coupling between two solutions of an SDE, and another one that relies on the differentiation of Kantorovitch dual formulation of optimal transport. We also provide more specific bounds in the case of the overdamped Langevin process, for which we are able to compare convergence to the invariant measure and sensitivity to the parameter.
Paper Structure (4 sections, 2 theorems, 50 equations)

This paper contains 4 sections, 2 theorems, 50 equations.

Table of Contents

  1. Background and motivations
  2. A first proof of Theorem 1.1. using Kantorovitch's dual formulation
  3. A second proof of Theorem 1.1. using a synchronous coupling
  4. Further results for the overdamped Langevin process

Key Result

Theorem 1.1

Let $p \in [2,+\infty)$, $a, a' \in \mathbb{R}^p$ and $\beta, \beta'>0$. There exist $C_{1,d,p}, C_{2,d,p} >0$ two explicit constants such that

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Example 1.1
  • Theorem 4.1
  • proof