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Feng-Yu Wang

Publications and source records attributed to Feng-Yu Wang.

At least 109 records · Page 6Linked to original sources
arXiv2016

Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Let $P_t$ be the (Neumann) diffusion semigroup $P_t$ generated by a weighted Laplacian on a complete connected Riemannian manifold $M$ without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant $\ll>0$ if and only if $$W_p(μ_1P_t, μ_2P_t)\le \e^{-\ll t} W_p (μ_1,μ_2),\ \ t\ge 0, p\ge 1 $$ holds for all probability measures $μ_1$ and $μ_2$ on $M$, where $W_p$ is the $L^p$ Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction $$W_p(μ_1P_t, μ_2P_t)\le c\e^{-\ll t} W_p (μ_1,μ_2),\ \ p\ge 1, t\ge 0$$ for some constants $c,\ll>0$ for a class of diffusion semigroups with negative curvature where the constant $c$ is essentially larger than $1$. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise.

math.PR↗
arXiv2016

Integrability Conditions for SDEs and Semi-Linear SPDEs

By using the local dimension-free Harnack inequality established on incomplete Riemannian manifolds, integrability conditions on the coefficients are presented for SDEs to imply the non-explosion of solutions as well as the existence, uniqueness and regularity estimates of invariant probability measures. These conditions include a class of drifts unbounded on compact domains such that the usual Lyapunov conditions can not be verified. The main results are extended to second order differential operators on Hilbert spaces and semi-linear SPDEs.

math.PR↗
arXiv2016

Integration by Parts Formula and Applications for SPDEs with Jumps

By using the Malliavin calculus and finite jump approximations, the Driver-type integration by parts formula is established for the semigroup associated to stochastic (partial) differential equations with noises containing a subordinate Brownian motion. As applications, the shift Harnack inequality and heat kernel estimates are derived. The main results are illustrated by SDEs driven by $å$-stable like processes.

math.PR↗
arXiv2015

Asymptotics of Sample Entropy Production Rate for Stochastic Differential Equations

By using the dimension-free Harnack inequality and the integration by parts formula for the associated diffusion semigroup, we prove the central limit theorem, the moderate deviation principle, and the logarithmic iteration law for the sample entropy production rate of stochastic differential equations with Lipschitz continuous and dissipative drifts.

math.PR↗
arXiv2015

Hypercontractivity for Functional Stochastic Partial Differential Equations

Explicit sufficient conditions on the hypercontractivity are presented for two classes of functional stochastic partial differential equations driven by, respectively, non-degenerate and degenerate Gaussian noises. Consequently, these conditions imply that the associated Markov semigroup is $L^2$-compact and exponentially convergent to the stationary distribution in entropy, variance and total variational norm. As the log-Sobolev inequality is invalid under the framework, we apply a criterion presented in the recent paper \cite{Wang14} using Harnack inequality, coupling property and Gaussian concentration property of the stationary distribution. To verify the concentration property, we prove a Fernique type inequality for infinite-dimensional Gaussian processes which might be interesting by itself.

math.PR↗
arXiv2015

Gradient estimates for SDEs Driven by Multiplicative Lévy Noise

Gradient estimates are derived, for the first time, for the semigroup associated to a class of stochastic differential equations driven by multiplicative Lévy noise. In particular, the estimates are sharp for $α$-stable type noises. To derive these estimates, a new derivative formula of Bismut-Elworthy-Li's type is established for the semigroup by using the Malliavin calculus and a finite-jump approximation argument.

math.PR↗
arXiv2015

Degenerate SDE with Hölder-Dini Drift and Non-Lipschitz Noise Coefficient

The existence-uniqueness and stability of strong solutions are proved for a class of degenerate stochastic differential equations, where the noise coeffcicient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e. the second component) and locally Hölder-Dini continuous of order $\ff 2 3$ in the first component. Moreover, the weak uniqueness is proved under weaker conditions on the noise coefficient. Furthermore, if the noise coefficient is $C^{1+\vv}$ for some $\vv>0$ and the drift is Hölder continuous of order $å\in (\ff 2 3,1)$ in the first component and order $\bb\in(0,1) $ in the second, the solution forms a $C^1$-stochastic diffeormorphism flow. To prove these results, we present some new characterizations of Hölder-Dini space by using the heat semigroup and slowly varying functions.

math.PR↗
arXiv2015

Perturbations of Functional Inequalities for Lévy Type Dirichlet Forms

Perturbations of super Poincaré and weak Poincaré inequalities for Lévy type Dirichlet forms are studied. When the range of jumps is finite our results are natural extensions to the corresponding ones derived earlier for diffusion processes; and we show that the study for the situation with infinite range of jumps is essentially different. Some examples are presented to illustrate the optimality of our results.

math.PR↗
arXiv2015

Functional Inequalities for Convolution Probability Measures

Let $μ$ and $ν$ be two probability measures on $\R^d$, where $μ(\d x)= \e^{-V(x)}\d x$ for some $V\in C^1(\R^d)$. Explicit sufficient conditions on $V$ and $ν$ are presented such that $μ*ν$ satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, the recent results on the log-Sobolev inequality derived in \cite{Z} for convolutions of the Gaussian measure and compactly supported probability measures are improved and extended.

math.PR↗
arXiv2015

Degenerate SDEs in Hilbert Spaces with Rough Drifts

The existence and uniqueness of mild solutions are proved for a class of degenerate stochastic differential equations on Hilbert spaces where the drift is Dini continuous in the component with noise and Hölder continuous of order larger than $\ff 2 3$ in the other component. In the finite-dimensional case the Dini continuity is further weakened. The main results are applied to solve second order stochastic systems driven by space-time white noises.

math.PR↗
arXiv2015

Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift

Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.

math.PR↗
arXiv2014

Harnack Inequality and Applications for Infinite-Dimensional GEM Processes

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in \cite{FW}. To prove the main results, explicit Harnack inequality and super Poincaré inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.

math.PR↗
arXiv2014

Exponential Convergence of Non-Linear Monotone SPDEs

For a Markov semigroup $P_t$ with invariant probability measure $μ$, a constant $\ll>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $μ$, if there exists a constant $C\in (0,\infty)$ such that $$ \sup_{μ(f^2)\le 1}\|P_tf-μ(f)\|_\infty \le C \e^{-\ll t},\ \ t\ge 1.$$ By using the coupling by change of measure in the line of [F.-Y. Wang, Ann. Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential convergence rate are derived for a class of non-linear monotone stochastic partial differential equations. The main result is illustrated by the stochastic porous medium equation and the stochastic $p$-Laplace equation respectively. Finally, the $V$-uniformly exponential convergence is investigated for stochastic fast-diffusion equations.

math.PR↗
arXiv2014

Hypercontractivity for Functional Stochastic Differential Equations

An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated to a class of functional stochastic differential equations. Consequently, the semigroup $P_t$ converges exponentially to its unique invariant probability measure $μ$ in entropy, $L^2(μ)$ and the totally variational norm, and it is compact in $L^2(μ)$ for large $t>0$. This provides a natural class of non-symmetric Markov semigroups which are compact for large time but non-compact for small time. A semi-linear model which may not satisfy this sufficient condition is also investigated.

math.PR↗
arXiv2014

Asymptotic Couplings by Reflection and Applications for Non-Linear Monotone SPDES

Asymptotic couplings by reflection are constructed for a class of non-linear monotone SPDES (stochastic partial differential equations). As applications, the gradient/Hölder estimates as well as the exponential convergence are derived for the associated Markov semigroup. The main results are illustrated by stochastic generalized porous media equations, stochastic $p$-Laplacian equations, and stochastic generalized fast-diffusion equations. We emphasize that the gradient estimate is studied at the first time for these equations.

math.PR↗
arXiv2014

Derivative Formulae and Poincaré Inequality for Kohn-Laplacian Type Semigroups

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator $L:=\ff 1 2 \sum_{i=1}^m X_i^2$ on $\R^{m+d}:= \R^m\times\R^d$ is investigated, where $$X_i(x,y)= \sum_{k=1}^m \si_{ki} \pp_{x_k} + \sum_{l=1}^d (A_l x)_i\pp_{y_l},\ \ (x,y)\in\R^{m+d}, 1\le i\le m$$ for $\si$ an invertible $m\times m$-matrix and $\{A_l\}_{1\le l\le d}$ some $m\times m$-matrices such that the Hörmander condition holds. We first establish Bismut-type and Driver-type derivative formulae with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.

math.PR↗
arXiv2014

Integration by Parts Formula and Shift Harnack Inequality for Stochastic Equations

A new coupling argument is introduced to establish Driver's integration by parts formula and shift Harnack inequality. Unlike known coupling methods where two marginal processes with different starting points are constructed to move together as soon as possible, for the new-type coupling the two marginal processes start from the same point but their difference is aimed to reach a fixed quantity at a given time. Besides the integration by parts formula, the new coupling method is also efficient to imply the shift Harnack inequality. Differently from known Harnack inequalities where the values of a reference function at different points are compared, in the shift Harnack inequality the reference function, rather than the initial point, is shifted. A number of applications of the integration by parts and shift Harnack inequality are presented. The general results are illustrated by some concrete models including the stochastic Hamiltonian system where the associated diffusion process can be highly degenerate, delayed SDEs, and semi-linear SPDEs.

math.PR↗