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Feng-Yu Wang

Publications and source records attributed to Feng-Yu Wang.

At least 73 records · Page 4Linked to original sources
arXiv2021

Wasserstein Convergence Rate for Empirical Measures on Noncompact Manifolds

Let $X_t$ be the (reflecting) diffusion process generated by $L:=Δ+\nabla V$ on a complete connected Riemannian manifold $M$ possibly with a boundary $\partial M$, where $V\in C^1(M)$ such that $μ(d x):= e^{V(x)}d x$ is a probability measure. We estimate the convergence rate for the empirical measure $μ_t:=\frac 1 t \int_0^t δ_{X_s\d s$ under the Wasserstein distance. As a typical example, when $M=\mathbb R^d$ and $V(x)= c_1- c_2 |x|^p$ for some constants $c_1\in \mathbb R, c_2>0$ and $p>1$, the explicit upper and lower bounds are present for the convergence rate, which are of sharp order when either $d<\frac{4(p-1)}p$ or $d\ge 4$ and $p\to\infty$.

math.PR↗
arXiv2021

Exponential Ergodicity for Non-Dissipative McKean-Vlasov SDEs

Under Lyapunov and monotone conditions, the exponential ergodicity in the induced Wasserstein quasi-distance is proved for a class of fully non-dissipative McKean-Vlasov SDEs, which strengthen some recent results established under dissipative conditions in long distance. Moreover, when the SDE is order-preserving, the exponential ergodicity is derived in the Wasserstein distance induced by one-dimensional increasing functions chosen according to the coefficients of the equation.

math.PR↗
arXiv2021

Derivative Formulas in Measure on Riemannian Manifolds

We characterise the link of derivatives in measure, which are introduced in [AKR,Card,ORS] respectively by different means, for functions on the space $\mathbb M$ of finite measures over a Riemannian manifold $M$. For a reasonable class of functions $f$, the extrinsic derivative $D^Ef$ coincides with the linear functional derivative $D^Ff$, the intrinsic derivative $D^If$ equals to the $L$-derivative $D^Lf$, and $$D^If(η)(x)= D^{L}f(η)(x)= \lim_{s\downarrow 0} \frac 1 s \nabla f(η+s δ_\cdot)(x) = \nabla \big\{D^E f (η)\big\}(x), \ \ (x,η)\in M\times\mathbb M,$$ where $\nabla$ is the gradient on $M$, $δ_x$ is the Dirac measure at $x$, and $$D^Ef(η)(x):= \lim\limits_{s\downarrow 0} \frac { f(η+s δ_x)-f(η)} s,\ \ x\in M$$ is the extrinsic derivative of $f$ at $η\in \mathbb M$. This gives a simple way to calculate the intrinsic or $L$-derivative, and is extended to functions of probability measures. %This provides a simple way to calculate the intrinsic/Lions derivative.

math.PR↗
arXiv2021

Bismut Formula for Lions Derivative of Distribution Dependent SDEs and Applications

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of $P_tf(μ):=\mathbb E f(X_t^μ)$, where $t>0,$ $ f $ is a bounded measurable function, and $X_t^μ$ solves a distribution dependent SDE with initial distribution $μ$. As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semigroup property and the invalidity of the formula $P_tf (μ)= \int P_tf(x)μ(d x)$, essential difficulties are overcome in the study.

math.PR↗
arXiv2021

Bismut Formula for Lions Derivative of Distribution-Path Dependent SDEs

To characterize the regularity of distribution-path dependent SDEs in the initial distribution which varies in the class of probability measures on the path space, we introduce the intrinsic and Lions derivatives for probability measures on Banach spaces, and prove the chain rule of the Lions derivative for the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is invalid, we establish the asymptotic Bismut formula. Both non-degenerate and degenerate noises are considered. The main results of this paper generalize and improve the corresponding ones derived recently in the literature for the classical SDEs with memory and McKean-Vlasov SDEs without memory.

math.PR↗
arXiv2021

Precise Limit in Wasserstein Distance for Conditional Empirical Measures of Dirichlet Diffusion Processes

Let $M$ be a $d$-dimensional connected compact Riemannian manifold with boundary $\partial M$, let $V\in C^2(M)$ such that $μ(dx):=e^{V(x)} d x$ 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 conditional empirical measure $μ_t^ν:= \mathbb E^ν\big(\frac 1 t \int_0^t δ_{X_s}d s\big|t<τ\big)$ for the diffusion process with initial distribution $ν$ such that $ν(\partial M)<1$. Then $$\lim_{t\to\infty} \big\{t\mathbb W_2(μ_t^ν,μ_0)\big\}^2 = \frac 1 {\{μ(ϕ_0)ν(ϕ_0)\}^2} \sum_{m=1}^\infty \frac{\{ν(ϕ_0)μ(ϕ_m)+ μ(ϕ_0) ν(ϕ_m)\}^2}{(λ_m-λ_0)^3},$$ where $ν(f):=\int_Mf {d} ν$ for a measure $ν$ and $f\in L^1(ν)$, $μ_0:=ϕ_0^2μ$, $\{ϕ_m\}_{m\ge 0}$ is the eigenbasis of $-L$ in $L^2(μ)$ with the Dirichlet boundary, $\{λ_m\}_{m\ge 0}$ are the corresponding Dirichlet eigenvalues, and $\mathbb W_2$ is the $L^2$-Wasserstein distance induced by the Riemannian metric.

math.PR↗
arXiv2021

Convergence in Wasserstein Distance for Empirical Measures of Semilinear SPDEs

The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.

math.PR↗
arXiv2020

Distribution Dependent Stochastic Differential Equations

Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs for short) have been intensively investigated. In this paper we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.

math.PR↗
arXiv2020

Linearization of Nonlinear Fokker-Planck Equations and Applications

We associate a coupled nonlinear Fokker-Planck equation on $\R^d$, i.e. with solution paths in $\scr P$, to a linear Fokker-Planck equation for probability measures on the product space $\R^d\times \scr P$, i.e. with solution paths in $\scr P(\R^d\times\scr P)$. We explicitly determine the corresponding linear Kolmogorov operator $\tilde{\mathbf{L}}_t$ using the natural tangent bundle over $\scr P$ with corresponding gradient operator $\nabla^\scr P$. Then it is proved that the diffusion process generated by $\tilde{\mathbf{L}}_t$ on $\R^d\times\scr P$ is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by $\tilde{\mathbf{L}}_t$ in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurebility conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of "Nemytskii-type". As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schrödinger type PDEs on $\R^d\times\scr P_2$, through the Feynman-Kac formula for the corresponding diffusion processes.

math.PR↗
arXiv2020

Image Dependent Conditional McKean-Vlasov SDEs for Measure-Valued Diffusion Processes

We consider a special class of mean field SDEs with common noise which depend on the image of the solution (i.e. the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games, the Feynman-Kac formula is established to solve Schrördinegr type PDEs on $\scr P_2$, and the ergodicity is proved for a class of measure-valued diffusion processes.

math.PR↗
arXiv2020

Nonlinear Fokker-Planck equations for Probability Measures on Path Space and Path-Distribution Dependent SDEs

By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker--Planck equations for probability measures $(μ_t)_{t \geq 0}$ on the path space $\mathcal C:=C([-r_0,0];\mathbb R^d),$ is analyzed: $$\partial_t μ(t)=L_{t,μ_t}^*μ_t,\ \ t\ge 0,$$ where $μ(t)$ is the image of $μ_t$ under the projection $\mathcal C\niξ\mapsto ξ(0)\in\mathbb R^d$, and $$L_{t,μ}(ξ):= \frac 1 2\sum_{i,j=1}^d a_{ij}(t,ξ,μ)\frac{\partial^2} {\partial_{ξ(0)_i} \partial_{ξ(0)_j}} +\sum_{i=1}^d b_i(t,ξ,μ)\frac{\partial}{\partial_{ξ(0)_i}},\ \ t\ge 0, ξ\in \mathcal C, μ\in \mathcal P^{\mathcal C}.$$ Under reasonable conditions on the coefficients $a_{ij}$ and $b_i$, the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.

math.PR↗
arXiv2020

Derivative Estimates on Distributions of McKean-Vlasov SDEs

By using the heat kernel parameter expansion with respect to the frozen SDEs, the intrinsic derivative is estimated for the law of Mckean-Vlasov SDEs with respect to the initial distribution. As an application, the total variation distance between the laws of two solutions is bounded by the Wasserstein distance for initial distributions. These extend some recent results proved for distribution-free noise by using the coupling method and Malliavin calculus.

math.PR↗
arXiv2020

McKean-Vlasov SDEs with Drifts Discontinuous under Wasserstein Distance

Existence and uniqueness are proved for Mckean-Vlasov type distribution dependent SDEs with singular drifts satisfying an integrability condition in space variable and the Lipschitz condition in distribution variable with respect to $W_0$ or $W_0+W_θ$ for some $θ\ge 1$, where $W_0$ is the total variation distance and $W_θ$ is the $L^θ$-Wasserstein distance. This improves some existing results where the drift is either locally bounded in the space variable or continuous in the distribution variable with respect to the Wasserstein distance.

math.PR↗
arXiv2020

Donsker-Varadhan Large Deviations for Path-Distribution Dependent SPDEs

As an important tool characterizing the long time behavior of Markov processes, the Donsker-Varadhan LDP (large deviation principle) does not directly apply to distribution dependent SDEs/SPDEs since the solutions are non-Markovian. We establish this type LDP for several different models of distribution dependent SDEs/SPDEs which may also with memories, by comparing the original equations with the corresponding distribution independent ones. As preparations, the existence, uniqueness and exponential convergence are also investigated for path-distribution dependent SPDEs which should be interesting by themselves.

math.PR↗
arXiv2020

Functional Inequalities for Weighted Gamma Distribution on the Space of Finite Measures

Let $\MM$ be the space of finite measures on a Locally compact Polish space, and let $\BG$ be the Gamma distribution on $\MM$ with intensity measure $ν\in \MM$. Let $\nn^{ext}$ be the extrinsic derivative with tangent bundle $T\MM= \cup_{η\in\MM} L^2(η)$, and let $\GA: T\MM\to T\MM$ be measurable such that $\GA_η$ is a positive definite linear operator on $L^2(η)$ for every $η\in \MM$. Moreover, for a measurable function $V$ on $\MM$, let $\d\BG^V= \e^V\d\BG$. We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form $$\EE_{\GA,V}(F,G):= \int_\MM \<\GA_η\nn^{ext}F(η), \nn^{ext}G(η)\>_{L^2(η)}\, \d\BG^V(η),$$ which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.

math.PR↗
arXiv2019

Spectral Gap for Measure-Valued Diffusion Processes

The spectral gap is estimated for measure-valued diffusion processes induced by the intrinsic/extrinsic derivatives on the space of finite measures over a Riemannian manifold. This provides explicit exponential convergence rate for these processes to approximate the Dirichlet and Gamma distributions arising from population genetics.

math.PR↗
arXiv2019

Limit Theorems for Additive Functionals of Path-Dependent SDEs

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

math.PR↗