arXiv subjects

Liying Mu

Publications and source records attributed to Liying Mu.

3 recordsLinked to original sources

arXiv2026

Well-posedness and Quantitative Stability for Degenerate Fokker--Planck Equations with Rough Coefficients

We establish quantitative stability and well-posedness for degenerate Fokker--Planck equations with Sobolev-regular and possibly fully degenerate diffusion coefficient. For two classes of rough drifts, stability estimates imply uniqueness, and existence is proved under suitable conditions. First, for anisotropic drifts with partial fractional Sobolev regularity, we prove both existence (as probability densities, essential for the probabilistic interpretation via superposition) and quantitative stability without imposing any growth conditions, covering the full range of fractional exponents and extending previous Lagrangian-flow results. Second, for drifts whose distributional gradients are given by singular integrals (e.g., Biot--Savart velocity fields from $L^1$ vorticity), we obtain stability estimates that allow the diffusion matrix to be time-space dependent and possibly degenerate, removing the prior restrictions of identity diffusion or bounded coefficients. Our results also recover the pure transport regime as a limiting case.

math.PR↗

arXiv2025

Littlewood--Paley--Stein Square Functions for the Fractional Discrete Laplacian on $\mathbb{Z}$

We investigate the boundedness of ``vertical'' Littlewood--Paley--Stein square functions for the nonlocal fractional discrete Laplacian on the lattice $\mathbb{Z}$, where the underlying graphs are not locally finite. When $q\in[2,\infty)$, we prove the $l^q$ boundedness of the square function by exploring the corresponding Markov jump process and applying the martingale inequality. When $q\in (1,2]$, we consider a modified version of the square function and prove its $l^q$ boundedness through a careful in on the generalized carré du champ operator. A counterexample is constructed to show that it is necessary to consider the modified version. Moreover, we extend the study to a class of nonlocal Schrödinger operators for $q\in (1,2]$.

math.PR↗

arXiv2025

Exponential Ergodicity in $\W_1$ for SDEs with Distribution Dependent Noise and Partially Dissipative Drifts

Being concerned with ergodicity of McKean--Vlasov SDEs, we establish a general result on exponential ergodicity in the $L^1$-Wasserstein distance. The result is successfully applied to non-degenerate and multiplicative Brownian motion cases, degenerate second order systems, and even the additive $α$-stable noise, where the coefficients before the noise are allowed to be distribution dependent and the drifts are only assumed to be partially dissipative. Our results considerably improve existing ones whose coefficients before the noise are distribution-free.

math.PR↗