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Feng-Yu Wang

Publications and source records attributed to Feng-Yu Wang.

At least 91 records · Page 5Linked to original sources
arXiv2018

Asymptotic Log-Harnack Inequality and Applications for Stochastic Systems of Infinite Memory

The asymptotic log-Harnack inequality is established for several different models of stochastic differential systems with infinite memory: non-degenerate SDEs, Neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate; uniqueness of the invariant probability measure; asymptotic gradient estimate and hence, asymptotically strong Feller property; and asymptotic irreducibilty.

math.PR↗
arXiv2018

Gradient Estimates on Dirichlet Eigenfunctions

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants $c\_1(D)$ and $c\_2(D)$ for a $d$-dimensional compact Riemannian manifold $D$ with boundary such that $c\_1(D)\sqrtλ\|ϕ\|\_\infty \le \|\nabla ϕ\|\_\infty\le c\_2(D)\sqrtλ \|ϕ\|\_\infty$ holds for any Dirichlet eigenfunction $ϕ$ of $-Δ$ with eigenvalue $λ$. In particular, when $D$ is convex with nonnegative Ricci curvature, this estimate holds for $c\_1(D)=\frac{1}{de}$ and $c\_2(D)=\sqrt{e}\left(\frac{\sqrt{2}}{\sqrtπ}+\frac{\sqrtπ}{4\sqrt{2}}\right)$. Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the second part of the paper.

math.PR↗
arXiv2018

Space-Distribution PDEs for Path Independent Additive Functionals of McKean-Vlasov SDEs

Let P2(Rd) be the space of probability measures on Rd with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space Rd*P2(Rd) equipped with the usual derivative in space variable and Lions derivative in distribution. These PDEs are solved by using probabilis- tic arguments developed from [2]. In particular, the path independence of the Girsanov transformation killing the drift term is identified with a nonlinear PDE on Rd*P2(Rd), which includes corresponding results derived earlier for the classical SDEs as special situations.

math.PR↗
arXiv2018

Ergodicity for Neutral Type SDEs with Infinite Length of Memory

In this paper, the weak Harris theorem developed in \cite{HMS11} is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.

math.PR↗
arXiv2018

Distribution Dependent SDEs with Singular Coefficients

Under integrability conditions on distribution dependent coefficients, existence and uniqueness are proved for McKean-Vlasov type SDEs with non-degenerate noise. When the coefficients are Dini continuous in the space variable, gradient estimates and Harnack type inequalities are derived. These generalize the corresponding results derived for classical SDEs, and are new in the distribution dependent setting.

math.PR↗
arXiv2018

Degenerate SDEs with Singular Drift and Applications to Heisenberg Groups

By using the ultracontractivity of a reference diffusion semigroup, Krylov's estimate is established for a class of degenerate SDEs with singular drifts, which leads to existence and pathwise uniqueness by means of Zvonkin's transformation. The main result is applied to singular SDEs on generalized Heisenberg groups.

math.PR↗
arXiv2018

Nash inequality for Diffusion Processes Associated with Dirichlet Distributions

For any $N\ge 2$ and $α=(α_1,\cdots, α_{N+1})\in (0,\infty)^{N+1}$, let $μ^{(N)}_α$ be the Dirichlet distribution with parameter $α$ on the set $Δ^{ (N)}:= \{ x \in [0,1]^N:\ \sum_{1\le i\le N}x_i \le 1 \}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form $${\scr E}_α^{(N)}(f,f):= \sum_{n=1}^N \int_{ Δ^{(N)}} \bigg(1-\sum_{1\le i\le N}x_i\bigg) x_n(\partial_n f)^2(x)\,μ^{(N)}_α(d x)$$ with Domain ${\scr D}({\scr E}_α^{(N)})$ being the closure of $C^1(Δ^{(N)})$. We prove the Nash inequality $$μ_α^{(N)}(f^2)\le C {\scr E}_α^{(N)}(f,f)^{\frac p{p+1} }μ_α^{(N)} (|f|)^{\frac 2 {p+1}},\ \ f\in {\scr D}({\scr E}_α^{(N)}), μ_α^{(N)}(f)=0$$ for some constant $C>0$ and $p= (α_{N+1}-1)^+ +\sum_{i=1}^N 1\lor (2α_i),$ where the constant $p$ is sharp when $\max_{1\le i\le N} α_i \le 1/2$ and $α_{N+1}\ge 1$. This Nash inequality also holds for the corresponding Fleming-Viot process.

math.PR↗
arXiv2018

Estimates for Invariant Probability Measures of Degenerate SPDEs with Singular and Path-Dependent Drifts

In terms of a nice reference probability measure, integrability conditions on the path-dependent drift are presented for (infinite-dimensional) degenerate PDEs to have regular positive solutions. To this end, the corresponding stochastic (partial) differential equations are proved to possess the weak existence and uniqueness of solutions, as well as the existence, uniqueness and entropy estimates of invariant probability measures. When the reference measure satisfies the log-Sobolev inequality, Sobolev estimates are derived for the density of invariant probability measures. Some results are new even for non-degenerate SDEs with path-independent drifts. The main results are applied to nonlinear functional SPDEs and degenerate functional SDEs/SPDEs.

math.PR↗
arXiv2017

Identifying Constant Curvature Manifolds, Einstein Manifolds, and Ricci Parallel Manifolds

We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified, respectively, with different integral-differential formulas and semigroup inequalities. Moreover, by using derivative and Hessian formulas for the heat semigroup $P_t$ developed from stochastic analysis, explicit Hessian estimates are derived on Einstein and Ricci parallel manifolds.

math.DG↗
arXiv2017

Order Preservation for Path-Distribution Dependent SDEs

Sufficient and necessary conditions are presented for the order preservation of path-distribution dependent SDEs. Differently from the corresponding study of distribution independent SDEs, to investigate the necessity of order preservation for the present model we need to construct a family of probability spaces in terms of the ordered pair of initial distributions.

math.PR↗
arXiv2017

Pointwise Characterizations of Curvature and Second Fundamental Form on Riemannian Manifolds

Let $M$ be a complete Riemannian manifold possibly with a boundary $\partial M$. For any $C^1$-vector field $Z$, by using gradient/functional inequalities of the (reflecting) diffusion process generated by $L:=Δ+Z$, pointwise characterizations are presented for the Bakry-Emery curvature of $L$ and the second fundamental form of $\partial M$ if exists. These extend and strengthen the recent results derived by A. Naber for the uniform norm $\|{\mathbf{Ric}}_Z\|_\infty$ on manifolds without boundary. A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first named author, such that the proofs are significantly simplified.

math.PR↗
arXiv2017

Isoperimetric Inequalities for Non-Local Dirichlet Forms

Let $(E,\F,μ)$ be a $\si$-finite measure space. For a non-negative symmetric measure $J(\d x, \d y):=J(x,y) \,μ(\d x)\,μ(\d y)$ on $E\times E,$ consider the quadratic form $$\E(f,f):= \frac{1}{2}\int_{E\times E} (f(x)-f(y))^2 \, J(\d x,\d y)$$ in $L^2(μ)$. We characterize the relationship between the isoperimetric inequality and the super Poincaré inequality associated with $\E$. In particular, sharp Orlicz-Sobolev type and Poincaré type isoperimetric inequalities are derived for stable-like Dirichlet forms on $\R^n$, which include the existing fractional isoperimetric inequality as a special example.

math.PR↗
arXiv2017

Higher Order Eigenvalues for Non-Local Schrödinger Operators

Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schrödinger operators by using the jump rate and the growth of the potential. For instance, let $L$ be the generator of a Lévy process with Lévy measure $ν(d z):= ρ(z) d z$ such that $ρ(z)=ρ(-z)$ and $$c_1 |z|^{-(d+α_1)}\le ρ(z)\le c_2|z|^{-(d+α_2)},\ \ |z|\le κ$$ for some constants $κ, c_1,c_2>0$ and $α_1,α_2\in (0,2),$ and let $c_3|x|^{θ_1} \le V(x)\le c_4|x|^{θ_2}$ for some constants $θ_1,θ_2, c_3,c_4>0$ and large $|x|$. Then the eigenvalues $λ_1\le λ_2\le\cdots λ_n\le \cdots $ of $-L+V$ satisfies the following two-side estimate: for any $p>1$, there exists a constant $C>1$ such that $$C n^{\frac{θ_2α_2}{d(θ_2+α_2)}}\ge λ_n \ge C^{-1} n^{\frac{θ_1α_1}{d(θ_1+α_1)}},\ \ n\ge 1.$$ When $α_1$ is variable, a better lower bound estimate is derived.

math-ph↗
arXiv2017

Distribution-Dependent SDEs for Landau Type Equations

The distribution-dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a non-linear PDE. Due to the distribution-dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a non-linear semigroup $P_t^*$ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the non-linear semigroup $P_t^*$ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.

math.PR↗
arXiv2017

Weak Poincaré Inequalities for Convergence Rate of Degenerate Diffusion Processes

For a contraction $C_0$-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincaré inequalities for the symmetric and anti-symmetric part of the generator. As applications, non-exponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.

math.PR↗
arXiv2017

Functional Inequalities for Non-Symmetric Stochastic Differential Equations

According to the theory of functional inequalities, a non-symmetric Markov semigroup has better properties than the corresponding symmetric one. For instance, there exist non-symmetric Markov semigroups which are hypercontractive (and thus converge exponentially in both $L^2$ and entropy), but the symmetric ones are even not ergodic. In this paper, we aim to search for reasonable conditions to ensure that a non-symmetric Markov semigroup and its symmetrization share the properties of exponential convergence, uniform integrability, hypercontractivity, and supercontractivity. Since in the symmetric case these properties are precisely characterized by functional inequalities of the Dirichlet form, the key point of the study is to prove these inequalities for non-symmetric Markov processes. SDEs driven by Brownian motion or Lévy jump process are investigated.

math.PR↗
arXiv2016

Hypercontractivity for Stochastic Hamiltonian Systems

The hypercontractivity is proved for the Markov semigroup associated to a class of finite/infinite dimensional stochastic Hamiltonian systems. Consequently, the Markov semigroup is exponentially convergent to the invariant probability measure in entropy (thus, also in $L^2$), and is compact for large time. Since the log-Sobolev inequality is invalid for the associated Dirichlet form, we introduce a general result on the hypercontractivity using the Harnack inequality with power. The main results are illustrated by concrete examples.

math.PR↗