arXiv subjects

Feng-Yu Wang

Publications and source records attributed to Feng-Yu Wang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Markov Processes and Stochastic Extrinsic Derivative Flows on the Space of Absolutely Continuous Measures

Let $E$ be the class of finite (resp. probability) measures absolutely continuous with respect to a $σ$-finite Radon measure on a Polish space. We present a criterion on the quasi-regularity of Dirichlet forms on $E$ in terms of upper bound conditions given by the uniform $(L^1+L^\infty)$-norm of the extrinsic derivative. As applications, we construct a class of general type Markov processes on $E$ via quasi-regular Dirichlet forms containing the diffusion, jump and killing terms. Moreover, stochastic extrinsic derivative flows on $E$ are studied by using quasi-regular Dirichlet forms, which in particular provide martingale solutions to SDEs on these two spaces, with drifts given by the extrinsic derivative of entropy functionals.

math.PR↗

arXiv2025

Bi-Coupling Method and Applications

By developing a new technique called the bi-coupling argument, we estimate the relative entropy between different diffusion processes in terms of the distances of initial distributions and drift-diffusion coefficients. As an application, the entropy-cost inequality is established for McKean-Vlasov SDEs with spatial-distribution dependent noise, which is open for a long time and has potential applications in optimal transport, information theory and mean field particle systems.

math.PR↗

arXiv2025

Log-Harnack Inequality and Bismut Formula for McKean-Vlasov SDEs with Singularities in all Variables

The log-Harnack inequality and Bismut formula are established for McKean-Vlasov SDEs with singularities in all (time, space, distribution) variables, where the drift satisfies an integrability condition in time-space, and the continuity in distribution may be weaker than Dini. The main results considerably improve the existing ones for the case where the drift is $L$-differentiable and Lipschitz continuous in distribution with respect to the 2-Wasserstein distance.

math.PR↗

arXiv2025

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

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↗

arXiv2024

Exponential Convergence in Entropy and Wasserstein Distance for McKean-Vlasov SDEs

The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(μ_t,μ_\infty)^2 +{\rm Ent}(μ_t|μ_\infty)\le c {\rm e}^{-λt} \min\big\{W_2(μ_0, μ_\infty)^2,{\rm Ent}(μ_0|μ_\infty)\big\},\ \ t\ge 1,$$ where $c,λ>0$ are constants, $μ_t$ is the distribution of the solution at time $t$, $μ_\infty$ is the unique invariant probability measure, ${\rm Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy.

math.PR↗

arXiv2024

Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes

For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E} [{\mathbb W}_p^q(μ_T,μ)])^{\frac{1}{q}} (T \to \infty)$ uniformly in $(p,q)\in [1,\infty) \times (0,\infty)$, where $μ_T$ is the empirical measure of the diffusion process, $μ$ is the unique invariant probability measure, and ${\mathbb W}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that ${\mathbb E} |T {\mathbb W}_2^2(μ_T,μ)-Ξ(T)|^q \to 0$ as $T\to\infty$ for any $q\ge 1$, where $Ξ(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.

math.PR↗

arXiv2024

Entropy Estimate for Degenerate SDEs with Applications to Nonlinear Kinetic Fokker-Planck Equations

The relative entropy for two different degenerate diffusion processes is estimated by using the Wasserstein distance of initial distributions and the difference between coefficients. As applications, the entropy cost inequality and exponential ergodicity in entropy are derived for distribution dependent stochastic Hamiltonian systems associated with nonlinear kinetic Fokker Planck equations.

math.PR↗

arXiv2024

Functional Inequalities for Brownian Motion on Riemannian Manifolds with Sticky-Reflecting Boundary Diffusion

We prove geometric upper bounds for the Poincaré and Logarithmic Sobolev constants for Brownian motion on manifolds with sticky reflecting boundary diffusion i.e. extended Wentzell-type boundary condition under general curvature assumptions on the manifold and its boundary. The method is based on an interpolation involving energy interactions between the boundary and the interior of the manifold. As side results we obtain explicit geometric bounds on the first nontrivial Steklov eigenvalue, for the norm of the boundary trace operator on Sobolev functions, and on the boundary trace logarithmic Sobolev constant. The case of Brownian motion with pure sticky reflection is also treated.

math.PR↗

arXiv2024

Ornstein-Uhlenbeck Type Processes on Wasserstein Space

Let $\mathcal P_2$ be the space of probability measures on $\R^d$ having finite second moment, and consider the Riemannian structure on $\mathcal P_2$ induced by the intrinsic derivative on the $L^2$-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein$-$Uhlenbeck (OU) type Dirichlet form on $\mathcal P_2$ whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is $L^2$-compact. Perturbations of the OU Dirichlet form are also studied.

math.PR↗

arXiv2023

Probability Distance Estimates Between Diffusion Processes and Applications to Singular McKean-Vlasov SDEs

The $L^k$-Wasserstein distance $\mathbb{W}_k (k\ge 1)$ and the probability distance $\mathbb{W}_ψ$ induced by a concave function $ψ$, are estimated between different diffusion processes with singular coefficients. As applications, the well-posedness, probability distance estimates and the log-Harnack inequality are derived for McKean-Vlasov SDEs with multiplicative distribution dependent noise, where the coefficients are singular in time-space variables and $(\mathbb{W}_k+\mathbb{W}_ψ)$-Lipschitz continuous in the distribution variable. This improves existing results derived in the literature under the $\mathbb{W}_k$-Lipschitz or derivative conditions in the distribution variable.

math.PR↗

arXiv2023

Hessian estimates for Dirichlet and Neumann eigenfunctions of Laplacian

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\fracλ{n}\,\|ϕ\|_{\infty} \leq \|{\rm Hess}\ ϕ\|_{\infty}\leq c(D)λ\,\|ϕ\|_{\infty}$ holds for any Dirichlet eigenfunction $ϕ$ of $-Δ$ with eigenvalue $λ$. Our results provide the sharp Hessian estimate $\|{\rm Hess}\ ϕ\|_{\infty}\lesssim λ^{\frac{n+3}{4}}$. Corresponding Hessian estimates for Neumann eigenfunctions are derived in the second part of the paper.

math.PR↗

arXiv2023

Exponential Ergodicity for Time-Periodic McKean-Vlasov SDEs

As extensions to the corresponding results derived for time homogeneous McKean- Vlasov SDEs, the exponential ergodicity is proved for time-periodic distribution dependent SDEs in three different situations: 1) in the quadratic Wasserstein distance and relative entropy for the dissipative case; 2) in the Wasserstein distance induced by a cost function for the partially dissipative case; and 3) in the weighted Wasserstein distance induced by a cost function and a Lyapunov function for the fully non-dissipative case. The main results are illustrated by time inhomogeneous granular media equations, and are extended to reflecting McKean-Vlasov SDEs in a convex domain.

math.PR↗

arXiv2023

Unadjusted Langevin Algorithms for SDEs with Hoelder Drift

Consider the following stochastic differential equation for $(X_t)_{t\ge 0}$ on $\mathbb R^d$ and its Euler-Maruyama (EM) approximation $(Y_{t_n})_{n\in \mathbb Z^+}$: \begin{align*} &d X_t=b( X_t) d t+σ(X_t) d B_t, \\ & Y_{t_{n+1}}=Y_{t_{n}}+η_{n+1} b(Y_{t_{n}})+σ(Y_{t_{n}})\left(B_{t_{n+1}}-B_{t_{n}}\right), \end{align*} where $b:\mathbb{R}^d \rightarrow \mathbb{R}^d,\ \ σ: \mathbb R^d \rightarrow \mathbb{R}^{d \times d}$ are measurable, $B_t$ is the $d$-dimensional Brownian motion, $t_0:=0,t_{n}:=\sum_{k=1}^{n} η_{k}$ for constants $η_k>0$ satisfying $\lim_{k \rightarrow \infty} η_k=0$ and $\sum_{k=1}^\inftyη_k =\infty$. Under (partial) dissipation conditions ensuring the ergodicity, we obtain explicit convergence rates of $\mathbb W_p(\mathscr{L}(Y_{t_n}), \mathscr{L}(X_{t_n}))+\mathbb W_p(\mathscr{L}(Y_{t_n}), μ)\rightarrow 0$ as $n\rightarrow \infty$, where $\mathbb W_p$ is the $L^p$-Wasserstein distance for certain $p\in [0,\infty)$, $\mathscr{L}(ξ)$ is the distribution of random variable $ξ$, and $μ$ is the unique invariant probability measure of $(X_t)_{t \ge 0}$. Comparing with the existing results where $b$ is at least $C^2$-smooth, our estimates apply to Hoelder continuous drift and can be sharp in several specific situations.

math.PR↗

arXiv2023

Density Dependent Singular Stochastic Differential Equations

The (strong and weak) well-posedness is proved for singular SDEs depending on the distribution density point-wisely and globally, where the drift satisfies a local integrability condition in time-spatial variables, and is Lipschitz continuous in the distribution density with respect to a local $L^k$-norm. Density dependent reflecting SDEs are also studied.

math.PR↗

arXiv2023

Regularities and Exponential Ergodicity in Entropy for SDEs Driven by Distribution Dependent Noise

As two crucial tools characterizing regularity properties of stochastic systems, the log-Harnack inequality and Bismut formula have been intensively studied for distribution dependent (McKean-Vlasov) SDEs. However, due to technical difficulties, existing results mainly focus on the case with distribution free noise. In this paper, we introduce a noise decomposition argument to establish the log-Harnack inequality and Bismut formula for SDEs with distribution dependent noise, in both non-degenerate and degenerate situations. As application, the exponential ergodicity in entropy is investigated.

math.PR↗

arXiv2023

A General Framework for Solving Singular SPDEs with Applications to Fluid Models Driven by Pseudo-differential Noise

In this paper we focus on nonlinear SPDEs with singularities included in both drift and noise coefficients, for which the Gelfand-triple argument developed for (local) monotone SPDEs turns out to be invalid. We propose a general framework of \emph{proper regularization} to solve such singular SPDEs. As applications, the (local and global) existence is presented for a broad class of fluid models driven by pseudo-differential noise of arbitrary order, which include the stochastic magnetohydrodynamics (hence Navier-Stokes/Euler) equations, stochastic Camassa-Holm type equations, stochastic aggregation-diffusion equation and stochastic surface quasi-geostrophic equation. Thus, some recent results derived in the literature are considerably extended in a unified way.

math.AP↗