arXiv subjects
Publications and source records attributed to Feng-Yu Wang.
The well-posedness and exponential ergodicity are proved for stochastic Hamiltonian systems containing a singular drift term which is locally integrable in the component with noise. As an application, the well-posedness and uniform exponential ergodicity are derived for a class of singular degenerated McKean-Vlasov SDEs.
By refining a recent result of Xie and Zhang, we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean-Vlasov SDEs with or without reflection. We also present the exponential ergodicity in the relative entropy and (weighted) Wasserstein distances for reflecting McKean-Vlasov SDEs under the dissipative and partially dissipative conditions respectively. The main results are illustrated by non-symmetric singular granular media equations.
The Bismut formula is established for the intrinsic derivative of singular McKean-Vlasov SDEs, where the noise coefficient belongs to a local Sobolev space, and the drift contains a locally integrable time-space term as well as a time-space-distribution term Lipschitz continuous in the space and distribution variables. The results are new also for classical SDEs.
For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on $\mathbb{R}^d$ or $\mathbb{T}^d:=\mathbb{R}^d/\mathbb{Z}^d,$ up to a time explicitly given by the initial datum and three constants coming from the upper bounds of the heat kernel and the Riesz transform. A mild well-posedness is also proved for $L^p$-bounded initial data. The blow-up is proved for both type solutions with finite maximal time.
By using the spectrum of the underlying symmetric diffusion operator, the convergence in $L^p$-Wasserstein distance $\mathbb W_p (p\ge 1)$ is characterized for the empirical measure $μ_t$ of non-symmetric subordinated diffusion processes in an abstract framework. The main results are applied to the subordinations of several typical models, which include the (reflecting) diffusion processes on compact manifolds, the conditional diffusion processes, the Wright-Fisher diffusion process, and hypoelliptic diffusion processes on {\bf SU}(2). In particular, for the (reflecting) diffusion processes on a compact Riemannian manifold with invariant probability measure $μ$: (1) the sharp limit of $t\mathbb W_2(μ_t,μ)^2$ is derived in $L^q(\mathbb P)$ for concrete $q\ge 1,$ which provides a precise characterization on the physical observation that a divergence-free perturbation accelerates the convergence in $\mathbb W_2$; (2) the sharp convergence rates are presented for $(\mathbb E[\mathbb W_{2p}(μ_t,μ)^{q}])^{\frac 1 q} (p,q\ge 1)$, where a critical phenomenon appears with the critical rate $t^{-1}\log t$ as $t\to\infty$.
In this paper, we study $L^p$-boundedness ($1<p\leq 2$) of the covariant Riesz transform on differential forms for a class of non-compact weighted Riemannian manifolds without assuming conditions on derivatives of curvature. We present in particular a local version of $L^p$-boundedness of Riesz transforms under two natural conditions, namely the curvature-dimension condition, and a lower bound on the Weitzenböck curvature endomorphism. As an application, the Calderón-Zygmund inequality for $1< p\leq 2$ on weighted manifolds is derived under the curvature-dimension condition as hypothesis.
For a complete connected Riemannian manifold $M$ let $V\in C^2(M)$ be such that $μ(d x)={\rm e}^{-V(x)} \mbox{vol}(d x)$ is a probability measure on $M$. Taking $μ$ as reference measure, we derive inequalities for probability measures on $M$ linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.
Let $M$ be a complete connected Riemannian manifold with boundary $\partial M$, and let $P_t$ be the Neumann semigroup generated by $\frac{ 1}{ 2} L$ where $L=Δ+Z$ for a $C^1$-vector field $Z$ on $M$. We establish Bismut type formulae for $LP_t f$ and ${\rm Hess}_{P_tf}$ and present estimates of these quantities under suitable curvature conditions. In case when $P_t$ is symmetric in $L^2(μ)$ for some probability measure $μ$, a new type of log-Sobolev inequality is established which links the relative entropy $H$, the Stein discrepancy $S$, and relative Fisher information $I$, generalizing the authors' recent work in the case without boundary.
To characterize nonlinear Dirichlet problems in an open domain, we investigate killed distribution dependent SDEs. By constructing the coupling by projection and using the Zvonkin/Girsanov transforms, the well-posedness is proved for three different situations: 1) monotone case with distribution dependent noise (possibly degenerate), 2) singular case with non-degenerate distribution dependent noise, and 3) singular case with non-degenerate distribution independent noise. In the first two cases the domain is $C^2$ smooth such that the Lipschitz continuity in initial distributions is also derived, and in the last case the domain is arbitrary.
To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.
By using distribution dependent Zvonkin's transforms and Malliavin calculus, the Bismut type formula is derived for the intrinisc/Lions derivatives of distribution dependent SDEs with singular drifts, which generalizes the corresponding results derived for classical SDEs and regular distribution dependent SDEs.
By using Zvonkin's transformation and a two-step fixed point argument in distributions, the well-posedness and regularity estimates are derived for singular McKean-Vlasov SDEs with distribution dependent noise, where the drift contains a term growing linearly in space and distribution and a locally integrable term independent of distribution, while the noise coefficient is weakly differentiable in space and Lipschitz continuous in distribution with respect to the sum of Wasserstein and weighted variation distances. The main results extend existing ones derived for noise coefficients either independent of distribution, or having nice linear functional derivatives in distribution. Singular reflecting SDEs with distribution dependent noise are also studied.
Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $μ({\rm d} x):={\rm e}^{V(x)}{\rm d} x$ is a probability measure, and let $X_t$ be the diffusion process generated by $L:=Δ+\nabla V$ with $τ:=\inf\{t\ge 0: X_t\in\partial M\}$. Consider the empirical measure $μ_t:=\frac 1 t \int_0^t δ_{X_s}{\rm d} s$ under the condition $t<τ$ for the diffusion process. If $d\le 3$, then for any initial distribution not fully supported on $\partial M$, \begin{align*} &c\sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2} \le \liminf_{t\to \infty} \inf_{T\ge t} \Big\{t {\mathbb E}\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big]\Big\} \\ &\le \limsup_{t\to \infty} \sup_{T\ge t} \Big\{ t \mathbb E\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big] \Big\}\le \sum_{m=1}^\infty \frac{2}{(λ_m-λ_0)^2}\end{align*} holds for some constant $c\in (0,1]$ with $c=1$ when $\partial M$ is convex, where $μ_0:= ϕ_0^2μ$ for the first Dirichet eigenfunction $ϕ_0$ of $L$, $\{λ_m\}_{m\ge 0}$ are the Dirichlet eigenvalues of $-L$ listed in the increasing order counting multiplicities, and the upper bound is finite if and only if $d\le 3$. When $d=4$, $\sup_{T\ge t} \mathbb E\big[\mathbb W_2(μ_t, μ_0)^2\big|T<τ\big] $ decays in the order $t^{-1}\log t$, while for $d\ge 5$ it behaves like $t^{-\frac 2 {d-2}}$, as $t\to\infty$.
Terahertz (THz) technology has been a great candidate for applications, including pharmaceutic analysis, chemical identification, and remote sensing and imaging due to its non-invasive and non-destructive properties. Among those applications, penetrating-type hyperspectral THz signals, which provide crucial material information, normally involve a noisy, complex mixture system. Additionally, the measured THz signals could be ill-conditioned due to the overlap of the material absorption peak in the measured bands. To address those issues, we consider penetrating-type signal mixtures and aim to develop a blind hyperspectral unmixing (HU) method without requiring any information from a prebuilt database. The proposed HYperspectral Penetrating-type Ellipsoidal ReconstructION (HYPERION) algorithm is unsupervised, not relying on collecting extensive data or sophisticated model training. Instead, it is developed based on elegant ellipsoidal geometry under a very mild requirement on data purity, whose excellent efficacy is experimentally demonstrated.
By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa--Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.
Let $M$ be a compact connected Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $μ(d x):=e^{V(x)}d x$ is a probability measure, and let $\{λ_i\}_{i\ge 1} $ be all non-trivial eigenvalues of $-L$ with Neumann boundary condition if the boundary exists. Then the empirical measures $\{μ_t\}_{t>0}$ of the diffusion process generated by $L$ (with reflecting boundary if the boundary exists) satisfy $$ \lim_{t\to \infty} \big\{t \mathbb E^x [W_2(μ_{t},μ)^2]\big\}= \sum_{i=1}^\infty\frac 2 {λ_i^2}\ \text{ uniformly\ in\ } x\in M,$$ where $\mathbb E^x$ denotes the expectation for the diffusion process starting at point $x$, $W_2$ is the $L^2$-Warsserstein distance induced by the Riemannian metric. The limit is finite if and only if $d\le 3$, and in this case we derive the following central limit theorem: $$\lim_{t\to\infty} \sup_{x\in M} \Big|\mathbb P^x(t W_2(μ_{t},μ)^2<a)- \mathbb P\Big( \sum_{k=1}^\infty \frac{2ξ_k^2}{λ_k^2}<a\Big)\Big|=0, \ \ a\ge 0,$$ where $\mathbb P^x$ is the probability with respect to $\mathbb E^x$, and $\{ξ_k\}_{k\ge 1}$ are i.i.d. standard Gaussian random variables. Moreover, when $d\ge 4$ we prove that the main order of $\mathbb E^x[W_2(μ_{t},μ)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Moreover, when $d\ge 4$ the main order of $\mathbb E^x[W_2(μ_{t},μ)^2]$ is $t^{-\frac 2 {d-2}}$ as $t\to\infty$. Finally, we establish the long-time large deviation principle for $\{W_2(μ_t,μ)^2\}_{t\ge 0}$ with a good rate function given by the information with respect to $μ$.
To characterize the Neumann problem for nonlinear Fokker-Planck equations, we investigate distribution dependent reflecting SDEs (DDRSDEs) in a domain. We first prove the well-posedness and establish functional inequalities for reflecting SDEs with singular drifts, then extend these results to DDRSDEs with singular or monotone coefficients, for which a general criterion deducing the well-posedness of DDRSDEs from that of reflecting SDEs is established.
Let $M$ be a connected compact Riemannian manifold possibly with a boundary, let $V\in C^2(M)$ such that $μ(\d x):=\e^{V(x)}\d x$ is a probability measure, where $\d x$ is the volume measure, and let $L=Δ+\nabla V$. The exact convergence rate in Wasserstein distance is derived for empirical measures of subordinations for the (reflecting) diffusion process generated by $L$.