arXiv subjects
Publications and source records attributed to Feng-Yu Wang.
By using local and global versions of Bismut type derivative formulas, gradient estimates are derived for the Neumann semigroup on a narrow strip. Applications to functional/cost inequalities and heat kernel estimates are presented. Since the narrow strip we consider is non-convex with zero injectivity radius, and does not satisfy the volume doubling condition, existing results in the literature do not apply.
Sufficient and necessary conditions are presented for the order-preservation of stochastic functional differential equations on $\R^d$ with non-Lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency of the conditions extends and improves some known comparison theorems derived recently for one-dimesional equations and multidimensional equations without delay, and the necessity is new even in these special situations.
By using the $Φ$-entropy inequality derived in \cite{Wu, Ch} for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump Lévy processes. The semigroup $Φ$-entropy inequality for SDEs driven by Poisson point processes as well as a sharp result on the existence of invariant probability measures are also presented.
By using the Malliavin calculus and finite-jump approximations, the Driver-type integration by parts formula is established for the semigroup associated to stochastic 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.
By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived.
By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semi-linear stochastic equations in Hilbert spaces.
Let $V\in C^2(\R^d)$ such that $μ_V(\d x):= \e^{-V(x)}\,\d x$ is a probability measure, and let $å\in (0,2)$. Explicit criteria are presented for the $å$-stable-like Dirichlet form $$\E_{å,V}(f,f):= \int_{\R^d\times\R^d} \ff{|f(x)-f(y)|^2}{|x-y|^{d+α}}\,\d y\,\e^{-V(x)}\,\d x$$ to satisfy Poincaré-type (i.e., Poincaré, weak Poincaré and super Poincaré) inequalities. As applications, sharp functional inequalities are derived for the Dirichlet form with $V$ having some typical growths. Finally, the main result of \cite{MRS} on the Poincaré inequality is strengthened
By solving a control problem and using Malliavin calculus, explicit derivative formula is derived for the semigroup $P_t$ generated by the Gruschin type operator on $\R^{m}\times \R^{d}:$ $$L (x,y)=\ff 1 2 \bigg\{\sum_{i=1}^m \pp_{x_i}^2 +\sum_{j,k=1}^d (\si(x)\si(x)^*)_{jk} \pp_{y_j}\pp_{y_k}\bigg\},\ \ (x,y)\in \R^m\times\R^d,$$ where $\si\in C^1(\R^m; \R^d\otimes\R^d)$ might be degenerate. In particular, if $\si(x)$ is comparable with $|x|^{l}I_{d\times d}$ for some $l\ge 1$ in the sense of (\ref{A4}), then for any $p>1$ there exists a constant $C_p>0$ such that $$|\nn P_t f(x,y)|\le \ff{C_p (P_t |f|^p)^{1/p}}{ß{t}\land ß{t(|x|^2+t)^l}},\ \ t>0, f\in \B_b(\R^{m+d}), (x,y)\in \R^{m+d},$$ which implies a new Harnack type inequality for the semigroup. A more general model is also investigated.
A non-negative Markovian solution is constructed for a class of stochastic generalized porous media equations with reflection. To this end, some regularity properties and a comparison theorem are proved for stochastic generalized porous media equations, which are interesting by themselves. Invariant probability measures and ergodicity of the solution are also investigated.
By using lower bound conditions of the Lévy measure w.r.t. a nice reference measure, the coupling and strong Feller properties are investigated for the Markov semigroup associated with a class of linear SDEs driven by (non-cylindrical) Lévy processes on a Banach space. Unlike in the finite-dimensional case where these properties have also been confirmed for Lévy processes without drift, in the infinite-dimensional setting the appearance of a drift term is essential to ensure the quasi-invariance of the process by shifting the initial data. Gradient estimates and exponential convergence are also investigated. The main results are illustrated by specific models on the Wiener space and separable Hilbert spaces.
By constructing a coupling with unbounded time-dependent drift, dimension-free Harnack inequalities are established for a large class of stochastic differential equations with multiplicative noise. These inequalities are applied to the study of heat kernel upper bound and contractivity properties of the semigroup. The main results are also extended to reflecting diffusion processes on Riemannian manifolds with nonconvex boundary.
Due to technical reasons, existing results concerning Harnack type inequalities for SPDEs with multiplicative noise apply only to the case where the coefficient in the noise term is an Hilbert-Schmidt perturbation of a fixed bounded operator. In this paper we investigate a class of semi-linear SPDEs with strongly multiplicative noise whose coefficient is even allowed to be unbounded which is thus no way to be Hilbert-Schmidt. Gradient estimates, log-Harnack inequality and applications are derived. Applications to stochastic reaction-diffusion equations driven by space-time white noise are presented.
By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the consequent $L^2$-gradient inequality, are proved to be equivalent to the pointwise curvature lower bound condition together with the convexity or absence of the boundary. Some applications of the log-Harnack inequality are also introduced.
By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two results on existence and uniqueness of solutions on an open domain are presented.
By constructing a coupling in two steps and using the Girsanov theorem under a regular conditional probability, the log-Harnack inequality is established for a large class of Gruschin type semigroups whose generator might be both degenerate and non-Lipschitzian.
By using Girsanov transformation and martingale representation, Talagrand-type transportation cost inequalities, with respect to both the uniform and the $L^2$ distances on the global free path space, are established for the segment process associated to a class of neutral functional stochastic differential equations. Neutral functional stochastic partial differential equations are also investigated.
Let $L$ be an elliptic differential operator on a complete connected Riemannian manifold $M$ such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let $L^{(å)}$ be the $å$-stable subordination of $L$ for $å\in (1,2).$ We found some classes $\mathbb K_å^{\gg,\bb} (\bb,\gg\in [0,å))$ of time-space functions containing the Kato class, such that for any measurable $b: [0,\infty)\times M\to TM$ and $c: [0,\infty)\times M\to M$ with $|b|, c\in \mathbb K_å^{1,1},$ the operator $$L_{b,c}^{(å)}(t,x):= L^{(å)}(x)+ +c(t,x),\ \ (t,x)\in [0,\infty)\times M$$ has a unique heat kernel $p_{b,c}^{(å)}(t,x;s,y), 0\le s 1$, where $\rr$ is the Riemannian distance. The estimate of $\nabla_yp^{(å)}_{b,c}$ and the Hölder continuity of $\nn_x p_{b,c}^{(å)}$ are also considered. The resulting estimates of the gradient and its Hölder continuity are new even in the standard case where $L=\DD$ on $\R^d$ and $b,c$ are time-independent.
By constructing successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker-Planck equations on $\R^m\times\R^{d}$. The main results are also applied to the study of gradient estimate, entropy/transportation-cost inequality and heat kernel inequalities.