Quantitative stability estimates for Fokker-Planck equations

TL;DR

This paper develops quantitative stability estimates for Fokker-Planck equations with irregular coefficients, using superposition principles to handle degenerate and non-degenerate cases, advancing understanding of solution behavior under weak regularity conditions.

Contribution

It introduces new stability estimates for Fokker-Planck equations with irregular coefficients, employing Trevisan's superposition principle for both degenerate and non-degenerate cases.

Findings

Established stability estimates for degenerate Fokker-Planck equations.

Derived bounds for solutions with weakly differentiable coefficients.

Extended stability analysis to non-degenerate cases satisfying Ladyzhenskaya--Prodi--Serrin condition.

Abstract

We consider the Fokker--Planck equations with irregular coefficients. Two different cases are treated: in the degenerate case, the coefficients are assumed to be weakly differentiable, while in the non-degenerate case the drift satisfies only the Ladyzhenskaya--Prodi--Serrin condition. Using Trevisan's superposition principle which represents the solution as the marginal of the solution to the martingale problem of the diffusion operator, we establish quantitative stability estimates for the solutions of Fokker--Planck equations.