arXiv subjects

Feng-Yu Wang

Publications and source records attributed to Feng-Yu Wang.

At least 19 recordsLinked to original sources

arXiv2026

Dimension-free Convergence Rate in Sliced Wasserstein Distance for Empirical Measures of Markov Processes

To derive dimension-free convergence rates of empirical measures for Markov processes on a Banach space, we adopt the sliced Wasserstein distance (SW distance) induced by a probability measure with full support on the unit ball of the dual space. This distance is topologically stronger than the convergence in finite-dimensional distributions, and is topologically equivalent to the Wasserstein distance when the Banach space is finite-dimensional. Under this distance, we derive dimension-free convergence rates for the empirical measures of ergodic Markov processes on $\BB$, which can be sharp as illustrated by concrete examples. The study provides an efficient way to simulate infinite-dimensional distributions using sample trajectories of Markov processes, so that the $``$curse of dimensionality" appearing to the classical Wasserstein distance is avoided. The main results apply to a broad class of infinite-dimensional models, and are illustrated by partially dissipative SPDEs in the end of the paper.

math.PR↗

arXiv2026

Stochastic Euler Equations with Pseudo-differential Noise: Continuous and Discontinuous Perturbations in Compressible and Incompressible Flows

We study stochastic Euler equations in compressible and incompressible regimes, on the whole space and the torus, driven by mixed multiplicative noise: continuous Stratonovich/It\^o components and a discontinuous Marcus component. The noise amplitudes are pseudo-differential operators including the transport operator. We develop a local-in-time theory of classical solutions, establishing existence, uniqueness, and a blow-up criterion. Discontinuous Marcus noise requires new analytical tools to control interactions between jump discontinuities and nonlocal operators. For compressible equations, we formulate a transformation-and-extension principle recovering the Makino variable and its symmetric quasilinear formulation. Using transformed variables, compatible nonlinear Sobolev estimates are developed to close high-order stochastic energy bounds. This accommodates broad physically relevant state equations, including piecewise $\gamma$-laws, Chaplygin laws, and the white dwarf pressure law. Many equations remain unexplored in multidimensional stochastic compressible settings, even under pure It\^o forcing. For the incompressible damped case, we identify damping--noise regimes guaranteeing global existence, uniform bounds, and decay. To study statistical behavior, we establish a novel existence criterion for invariant probability measures tailored to Markov semigroups satisfying a \emph{restricted Feller property under mismatched metrics}. Bypassing single-topology Feller continuity robustly extends the Krylov--Bogoliubov theory. We use this to construct invariant measures for singular stochastic evolution systems in Hilbert spaces. For mixed multiplicative noise and $d\ge 2$, we prove existence of invariant measures for stochastic damped Euler equations on $\mathbb{T}^d$ under moderate damping--noise, and uniqueness under strong damping--noise on $\mathbb{T}^d$ and $\mathbb{R}^d$.

math.PR↗

arXiv2026

Asymptotic Log-Harnack Inequality for Degenerate SPDEs with Reflection

By constructing a suitable coupling by change of measures, the asymptotic log- Harnack inequality is established for a class of degenerate SPDEs with reflection. This inequality implies the asymptotic heat kernel estimate, the uniqueness of the invariant probability measure, the asymptotic gradient estimate (hence, asymptotically strong Feller property), and the asymptotic irreducibility. As application, the main result is illustrated by d-dimensional degenerate stochastic Navie-Stokes equations with reflection, where the dissipative operator is the Dirichlet Laplacian with a power \theta \geq 1 \vee \frac{d+2}{4}, which includes the Laplacian when d \geq 2.

math.PR↗

arXiv2026

McKean-Vlasov SDEs with Local Distributional Interactions: Well-Posedness and Entropy-Cost Estimates

We study McKean-Vlasov SDEs with interaction kernels in $\tt W^{-\dd,k},$ the local negative Sobolev space on $\R^d$ with indexes $\dd \in [0,\infty)$ and $k\in [1,\infty].$ We derive the local well-posedness for any singular indexes $(\dd,k)\in [0,\infty)\times [1,\infty],$ and prove the global well-posedness for any initial distributions provided $\dd+\ff d k<1$. Moreover, the relative entropy and the $\|\cdot\|_{\dd,k*}$-distance induced by $ \tt W^{-\dd,k}$ are estimated for the time-marginal distributions of solutions by using the Wasserstein distance of initial distributions, which describe the regularity of the solution in initial distribution. In particular, the main results apply to Nemytskii-type SDEs which depend on higher order derivatives of the density functions, as well as McKean-Vlasov SDEs with interactions more singular than Riesz kernels.

math.PR↗

arXiv2026

Regularity Estimates for Singular Density Dependent SDEs

Consider the density dependent (i.e. Nemytskii-type) SDEs on $\mathbb R^d$, where the drift $b_t(x,\rho(x),\rho)$ is locally integrable in $(t,x)\in [0,\infty)\times \mathbb R^d$ and may be singular in the distribution density function $\rho$. The relative/Renyi entropies between two time-marginal distributions are estimated by using the Wasserstein distance of initial distributions. When $d=1$ and $b_t$ decays at $t=0$ with rate $t^{\frac 1 2+}$, our the relative entropy estimate coincides with the classical entropy-cost inequality for elliptic diffusion processes. To estimate the Renyi entropy, a refined Khasminskii estimate is presented for singular SDEs which may be interesting by itself.

math.PR↗

arXiv2025

$L^p$-Boundedness of the Covariant Riesz Transform on Differential Forms for $p>2$

We establish the $L^p$-boundedness, for $p>2$, of the covariant Riesz transform $\nabla(\Delta_\mu^{(k)}+\sigma)^{-1/2} $ on differential forms over a class of complete weighted Riemannian manifolds. The proof is based on an heat-kernel criterion involving local volume doubling, heat kernel upper estimates, Kato-type curvature control, and gradient bounds for the heat semigroup on forms. Under curvature-dimension assumptions and Kato-type curvature bounds, this criterion applies and yields boundedness for all sufficiently large $\sigma$. In particular, in the unweighted case, the result confirms a conjecture of Baumgarth, Devyver and G\"uneysu~\cite{BDG-23}. As an application, we obtain Calder\'on--Zygmund inequalities for $p>2$ on weighted manifolds, which extends the recent work \cite{CCT} on manifolds without weight.

math.DG↗

arXiv2025

Super and Weak Poincar\'e Inequalities for Sticky-Reflected Diffusion Processes

As a continuation to \cite{MRW} where the Poincar\'e and log-Sobolev inequalities were studied for the sticky-reflected Brownian motion on Riemannian manifolds with boundary, this paper establishes the super and weak Poincar\'e inequalities for more general sticky-reflected diffusion processes. As applications, the convergence rate and uniform integrability of the associated diffusion semigroups are characterized. The main results are illustrated by concrete examples.

math.PR↗

arXiv2025

Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions

Let $k\in (d,\infty]$ and consider the $k*$-distance $$\|\mu-\nu\|_{k*}:= \sup\Big\{|\mu(f)-\nu(f)|:\ f\in\B_b(\R^d),\ \|f\|_{\tt L^k}:=\sup_{x\in \R^d}\|1_{B(x,1)}f\|_{L^k}\le 1\Big\}$$ between probability measures on $\R^d$. The exponential ergodicity in $1$-Wasserstein and $k*$ distances is derived for a class of McKean-Vlasov SDEs with small singular interactions measured by $\|\cdot\|_{k*}.$ Moreover, the exponential ergodicity in $2$-Wasserstein distance and relative entropy is derived when the interaction term is given by $$b^{(0)}(x,\mu) :=\int_{\R^d}h(x-y)\mu(\d y)$$ for some measurable function $h:\R^d\to\R^d$ with small $\|h\|_{\tt L^k}$.

math.PR↗

arXiv2025

Distribution Dependent Birth-Death Processes: $\mathbb{W}_p$-Estimate, Ergodicity and Propagation of Chaos

For a class of time inhomogenous distribution dependent birth-death processes, we derive the well-posedness, $\mathbb{W}_p$-estimate, exponential ergodicity, and uniform in time propagation of chaos. These extend the corresponding results derived for distribution dependent SDEs and mean field particle systems. As preparation, a criterion on the well-posedness of inhomogenous jump process is presented in the end of the paper, which should be interesting by itself.

math.PR↗

arXiv2025

Strong Feller Regularisation of 1-d Nonlinear Transport by Reflected Ornstein-Uhlenbeck Noise

We consider equations of nonlinear transport on the circle with regular self interactions appearing in aggregation models and deterministic mean field dynamics. We introduce a random perturbation of such systems through a stochastic orientation preserving flow, which is given as an integrated infinite dimensional periodic Ornstein- Uhlenbeck process with reflection. As our main result we show that the induced stochastic dynamics yields a measure valued Markov process on a class of regular measures. Moreover, we show that this process is strong Feller in the corresponding topology. This is interpreted as a qualitative regularisation by noise phenomenon.

math.PR↗

arXiv2025

Bismut Formula and Gradient Estimates for Dirichlet Semigroups with Application to Singular Killed DDSDEs

By establishing a local version of Bismut formula for Dirichlet semigroups on a regular domain, gradient estimates are derived for killed SDEs with singular drifts. As an application, the total variation distance between two solutions of killed DDSDEs is bounded above by the truncated $1$-Wasserstein distance of initial distributions, in the regular and singular cases respectively.

math.PR↗

arXiv2025

Stochastic intrinsic gradient flows on the Wasserstein space

We construct stochastic gradient flows on the $2$-Wasserstein space $\mathcal P_2$ over $\mathbb R^d$ for energy functionals of the type $W_F(\rho d x)=\int_{\mathbb R^d}F(x,\rho(x))d x$. The functions $F$ and $\partial_2 F$ are assumed to be locally Lipschitz on $\mathbb R^d\times (0,\infty)$. This includes the relevant examples of $W_F$ as the entropy functional or more generally the Lyapunov function of generalized porous media equations. First we define a class of Gaussian-based measures $\Lambda$ on $\mathcal P_2$ together with a corresponding class of symmetric Markov processes ${(R_t)}_{t\geq 0}$. Next, using Dirichlet form techniques we perform stochastic quantization for the perturbations of these objects which result from multiplying such a measure $\Lambda$ by a density proportional to $e^{-W_F}$. Finally we show that the intrinsic gradient $DW_F(\mu)$ is defined for $\Lambda$-a.e. $\mu$ and that the Gaussian-based reference measure $\Lambda$ can be chosen in such way that the distorted process ${(\mu_t)}_{t\geq 0}$ is a martingale solution for the equation $d\mu_t=-DW_F(\mu_t) d t+d R_t$, $t\geq 0$.

math.PR↗

arXiv2025

Entropy-Cost Inequalities for McKean-Vlasov SDEs with Singular Interactions

For a class of McKean-Vlasov stochastic differential equations with singular interactions, which include the Coulomb/Riesz/Biot-Savart kernels as typical examples (Examples 2.1 and 2.2), we derive the well-posedness and regularity estimates by establishing the entropy-cost inequality. To measure the singularity of interactions, we introduce a new probability distance induced by local integrable functions, and estimate this distance for the time-marginal laws of solutions by using the Wasserstein distance of initial distributions. A key point of the study is to characterize the path space of time-marginal distributions for the solutions, by using local hyperbound estimates on diffusion semigroups.

math.PR↗

arXiv2025

Path-Distribution Dependent SDEs: Well-Posedness and Asymptotic Log-Harnack Inequality

We consider stochastic differential equations on $\mathbb R^d$ with coefficients depending on the path and distribution for the whole history. Under a local integrability condition on the time-spatial singular drift, the well-posedness and Lipschitz continuity in initial values are proved, which is new even in the distribution independent case. Moreover, under a monotone condition, the asymptotic log-Harnack inequality is established, which extends the corresponding result of [5] derived in the distribution independent case.

math.PR↗

arXiv2024

Improving Numerical Error Bounds Near Sharp Interface Limit for Stochastic Reaction-Diffusion Equations

In the study of geometric surface evolutions, stochastic reaction-diffusion equation provides a powerful tool for capturing and simulating complex dynamics. A critical challenge in this area is developing numerical approximations that exhibit error bounds with polynomial dependence on $\vv^{-1}$, where the small parameter $\vv>0$ represents the diffuse interface thickness. The existence of such bounds for fully discrete approximations of stochastic reaction-diffusion equations remains unclear in the literature. In this work, we address this challenge by leveraging the asymptotic log-Harnack inequality to overcome the exponential growth of $\vv^{-1}$. Furthermore, we establish the numerical weak error bounds under the truncated Wasserstein distance for the spectral Galerkin method and a fully discrete tamed Euler scheme, with explicit polynomial dependence on $\vv^{-1}$.

math.NA↗

arXiv2024

Asymptotics in Wasserstein Distance for Empirical Measures of Markov Processes

In this paper we introduce some recent progresses on the convergence rate in Wasserstein distance for empirical measures of Markov processes. For diffusion processes on compact manifolds possibly with reflecting or killing boundary conditions, the sharp convergence rate as well as renormalization limits are presented in terms of the dimension of the manifold and the spectrum of the generator. For general ergodic Markov processes, explicit estimates are presented for the convergence rate by using a nice reference diffusion process, which are illustrated by some typical examples. Finally, some techniques are introduced to estimate the Wasserstein distance of empirical measures.

math.PR↗

arXiv2024

Probability Versions of Li-Yau Type Inequalities and Applications

By using stochastic analysis, two probability versions of Li-Yau type inequalities are established for diffusion semigroups on a manifold possibly with (non-convex) boundary. The inequalities are explicitly given by the Bakry-Emery curvature-dimension, as well as the lower bound of the second fundamental form if the boundary exists. As applications, a number of global and local estimates are presented, which extend or improve existing ones derived for manifolds without boundary. Compared with the maximum principle technique developed in the literature, the probabilistic argument we used is more straightforward and hence considerably simpler.

math.PR↗

arXiv2024

Wasserstein asymptotics for empirical measures of diffusions on four dimensional closed manifolds

We identify the leading term in the asymptotics of the quadratic Wasserstein distance between the invariant measure and empirical measures for diffusion processes on closed weighted four-dimensional Riemannian manifolds. Unlike results in lower dimensions, our analysis shows that this term depends solely on the Riemannian volume of the manifold, remaining unaffected by the potential and vector field in the diffusion generator.

math.PR↗