Chang-Song Deng, Xing Huang
Using coupling by change of measure and an approximation technique, Wang's Harnack inequalities are established for a class of functional SDEs driven by subordinate Brownian motions. The results cover the corresponding ones in the case without delay.
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Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power $α$ satisfies a dimension-free Harnack inequality provided $α\in(1/2, 1)$, and it satisfies the log-Harnack inequality for all $α\in (0,1).$ Some infinite-dimensional examples are also presented.
The existence and uniqueness are established for McKean-Vlasov SDEs driven by Lévy processes. By using an approximation technique and coupling by change of measures, Harnack inequalities are investigated for McKean-Vlasov SDEs driven by subordinate Brownian motions.
We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded $κ$-fat open set (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded $κ$-fat open sets with respect to these processes with their Euclidean boundary.
In this paper, we consider a large class of subordinate random walks $X$ on integer lattice $\mathbb{Z}^d$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions.
In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of $κ$-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.
We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting $0$, when $0$ is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolve…
In this paper, the coupling by change of measure is constructed for a class of SDEs with integrable drift and additive noise, from which the Harnack and shift Harnack inequalities are derived. Finally, as applications, the gradient estimate, the regularity of the heat kernel and the distribution properties of the associated transition probability are also obtained. The important tool is Krylov's estimate.
The Harnack and log Harnack inequalities for stochastic differential equation driven by $G$-Brownian motion with multiplicative noise are derived by means of coupling by change of mesure. All of the above results extend the existing ones in the linear expectation setting. Moreover, the gradient estimate generalize the nonlinear results appeared in [11].
By constructing successful couplings, the derivative formula, gradient estimates and Harnack inequalities are established for the semigroup associated with a class of degenerate functional stochastic differential equations.
We establish Harnack inequality and shift Harnack inequality for stochastic differential equation driven by $G$-Brownian motion. As applications, the uniqueness of invariant linear expectations and estimates on the $\sup$-kernel are investigated, where the $\sup$-kernel is introduced in this paper for the first time.
In this paper, we consider transient subordinate Brownian motion X in R^d, d \geq 1, where the Laplace exponent ϕof the corresponding subordinator satisfies some mild conditions. The scaleinvariant Harnack inequality is proved for X. We first give new forms of asymptotical properties of the Levy and potential density of the subordinator near zero. Using these results we find asymptotics of the Levy density and potential density of X near the origin, which is essential to our approach. The examples which are covered by our results include geometric stable processes and relativistic geometric st…
By using the coupling argument, we establish the Harnack and log-Harnack inequalites for stochastic differential equations with non-Lipschitz drifts and driven by additive anisotropic subordinated Brownian motions (in particular, cylindrical $α$-stable processes). Moreover, the gradient estimate is also derived when the drift is Lipschitz continuous.
In this paper we consider a class of stochastic differential equations driven by subordinate Brownian motion with Markovian switching. We use Malliavin calculus to study the smoothness of the density for the solution under uniform Hörmander's type condition.
We use a coupling method for functional stochastic differential equations with bounded memory to establish an analogue of Wang's dimension-free Harnack inequality \cite{MR1481127}. The strong Feller property for the corresponding segment process is also obtained.
We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.
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.
The well-posedness and regularity estimates in initial distributions are derived for singular McKean-Vlasov SDEs, where the drift contains a locally standard integrable term and a superlinear term in the spatial variable, and is Lipchitz continuous in the distribution variable with respect to a weighted variation distance. When the superlinear term is strengthened to be Lipschitz continuous, Wangs Harnack inequality is established. These results are new also for the classical Ito SDEs where the coefficients are distribution independent.
By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of \cite{W97} is also investigated.
The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers \cite{Flandoli, Zhang11}. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly Hölder continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric s…
In this paper, the Harnack inequalities for $G$-SDEs with degenerate noise are derived by method of coupling by change of measure. Moreover, the gradient estimate for the associated nonlinear semigroup $\bar{P}_t$ $$|\nabla \bar{P}_t f|\leq c(p,t)(\bar{P}_t |f|^p)^{\frac{1}{p}}, \ \ p>1, t>0$$ is also obtained for bounded and continuous function $f$. As an application of Harnack inequality, we prove the weak existence of degenerate $G$-SDEs under some integrable conditions. Finally, an example is presented. All of the above results extends the existed results in the linear expectation setting.
Asymptotic log-Harnack inequality for path-distribution dependent SDEs with infinite memory and Dini drift
针对具有无限记忆和Dini漂移的路径分布依赖随机微分方程，建立了渐近对数哈纳克不等式，该结果在分布无关情形下同样新颖。
We show that shift Harnack type inequalities (in the sense of F.-Y.~Wang \cite{Wan14}) are preserved under Bochner's subordination. The proofs are based on two types of moment estimates for subordinators. As a by-product we establish moment estimates for general Lévy processes.
In this paper, we consider a large class of subordinate Brownian motions $X$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss how such conditions govern the behavior of the subordinator and the corresponding subordinate Brownian motion for both large and small time and space. Then we establish a global uniform boundary Harnack principle in (unbounded) open sets for the subordinate Brownian motion. When the open set satisfies the interior and exterior ball conditions with radius $R >…
Log-Harnack Inequality and Bismut Formula for McKean-Vlasov SDEs with Singularities in all Variables
本文在漂移项于时间-空间可积且分布连续性弱于 Dini 的条件下，为所有变量均具有奇异性的 McKean-Vlasov 随机微分方程建立了 log-Harnack 不等式和 Bismut 公式，显著推广了现有结果。
Path-Distribution Dependent SDEs: Well-Posedness and Asymptotic Log-Harnack Inequality
本文研究了系数依赖历史路径与分布的随机微分方程，在局部可积条件下证明了适定性与初始值的Lipschitz连续性，并在单调条件下建立了渐近对数-Harnack不等式，推广了分布无关情形的已有结果。
In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in $C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bo…
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.
Let $W^D$ be a killed Brownian motion in a domain $D\subset {\mathbb R}^d$ and $S$ an independent subordinator with Laplace exponent $ϕ$. The process $Y^D$ defined by $Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator $-ϕ(-Δ|_D)$, where $Δ|_D$ is the Dirichlet Laplacian. In this paper we study the potential theory of $Y^D$ under a weak scaling condition on the derivative of $ϕ$. We first show that non-negative harmonic functions of $Y^D$ satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale inva…
Let $Z$ be a subordinate Brownian motion in ${\mathbb R}^d$, $d\ge 2$, via a subordinator with Laplace exponent $ϕ$. We kill the process $Z$ upon exiting a bounded open set $D\subset {\mathbb R}^d$ to obtain the killed process $Z^D$, and then we subordinate the process $Z^D$ by a subordinator with Laplace exponent $ψ$. The resulting process is denoted by $Y^D$. Both $ϕ$ and $ψ$ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of $Y^D$, in particular the boundary theory. First, in case that $D$ is a $κ$-fat bounded open set, we show that the Harn…
A subordinate Brownian motion $X$ is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $ϕ$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open…
Regularities and Exponential Ergodicity in Entropy for SDEs Driven by Distribution Dependent Noise
本文通过噪声分解方法，为分布依赖噪声驱动的SDE在非退化与退化情形下建立log-Harnack不等式和Bismut公式，并由此研究熵的指数遍历性。
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…
Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications
本文通过测度变换耦合为条件McKean-Vlasov SDE建立log-Harnack不等式，并导出带共同噪声的平均场粒子系统的相对熵定量条件混沌传播，结果推广至随机Hamilton系统。
Applying Zvonkin's transform, the exponential convergence in Wasserstein distance for a class of functional SDEs with Hölder continuous drift is obtained. This combining with log-Harnack inequality implies the same convergence in the sense of entropy, which also yields the convergence in total variation norm by Pinsker's inequality.
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.
In this paper we prove a derivative formula of Bismut-Elworthy-Li's type as well as gradient estimate for stochastic differential equations driven by $α$-stable noises, where $α\in(0,2)$. As an application, the strong Feller property for stochastic partial differential equations driven by subordinated cylindrical Brownian motions is presented.
For degenerate stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H>1/2$, the derivative formulas are established by using Malliavin calculus and coupling method, respectively. Furthermore, we find some relation between these two approaches. As applications, the (log) Harnack inequalities and the hyperbounded property are presented.
In this paper, Wang's log-Harnack inequality and exponential ergodicity are derived for two types of distribution dependent SDEs: one is the CKLS model, where the diffusion coefficient is a power function of order $θ$ with $θ\in[\frac{1}{2},1)$; the other one is Vasicek model, where the diffusion coefficient only depends on distribution. Both models in the distribution independent case can be used to characterize the interest rate in mathematical finance.
Boundary Harnack principle for non-local operators on metric measure spaces
本文给出度量测度空间上 Hunt 过程尺度不变边界 Harnack 不等式成立的充要条件，并证明从属布朗运动在满足内锥条件的 Lipschitz 区域中 BHP 成立，而在 Lipschitz 常数较大的截断圆锥中不成立。
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.
In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, using the decoupled method, some regularity such as relative entropy and Sobolev's estimate of invariant probability measure are proved. Finally, by Banach's fixed theorem, the existence and uniqueness of invariant probability measure for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian system with integrable drifts are obtained.
A subordinate Brownian motion is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is $-ϕ(-Δ)$, where $ϕ$ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with $ϕ$ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of indepe…
This work establishes the weak convergence of Euler-Maruyama's approximation for stochastic differential equations (SDEs) with singular drifts under the integrability condition in lieu of the widely used growth condition. This method is based on a skillful application of the dimension-free Harnack inequality. Moreover, when the drifts satisfy certain regularity conditions, the convergence rate is estimated. This method is also applicable when the diffusion coefficients are degenerate. A stochastic damping Hamiltonian system is studied as an illustrative example.
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 o…
Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs
针对部分耗散McKean-Vlasov SDEs在L²-瓦瑟斯坦距离和相对熵下指数遍历性问题，通过建立一致对数-索伯列夫不等式并结合对数-哈纳克不等式与塔拉格兰德不等式，解决该问题并扩展到退化扩散情形。
Coupling Methods and Applications on Path Dependent McKean-Vlasov SDEs
本文针对路径依赖McKean-Vlasov SDEs，运用条件测度变换耦合、渐近反射耦合等方法，推导了不同耗散条件下的指数收缩性及对数Harnack不等式，并得到路径依赖平均场粒子系统的时间一致混沌传播结果。
Regularities for distribution dependent SDEs with fractional noises
本文研究了由两个独立分数阶噪声驱动的分布依赖型 SDE 的正则性，利用向后耦合论证与分数阶导数算子分析，在非退化与退化情形下建立了 Lions 导数的 log-Harnack 不等式与 Bismut 公式。
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
In this work, by using the Malliavin calculus, under Hörmander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker-Planck (nonlocal) operator: \[\mathscr{L}^{(α)}_b:=Δ^{α/2}_{\mathrm{v}}+\mathrm {v}\cdot \nabla_x+b(x,\mathrm{v})\cdot \nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^d,…
One-sided measure theoretic elliptic operators and applications to SDEs driven by Gaussian white noise with atomic intensity
本文在一维环面上定义单边测度论椭圆算子，引入 Sobolev 型能量空间并构造 W-Brownian 桥与 W-Brownian 运动，进而证明相关确定性和随机微分方程解的存在唯一性。
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