Jianhai Bao, Feng-Yu Wang, Chenggui Yuan
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.
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By refining a recent result of Xie and Zhang, we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean-Vlasov SDEs with or without reflection. We also present the exponential ergodicity in the relative entropy and (weighted) Wasserstein distances for reflecting McKean-Vlasov SDEs under the dissipative and partially dissipative conditions respectively. The main results are illustrated by non-symmetric singular granula…
In this paper we show irreducibility and the strong Feller property for transition probabilities of stochastic differential equations with jumps and monotone coefficients. Thus, exponential ergodicity and the spectral gap for the corresponding transition semigroups are obtained.
Singular Degenerate SDEs: Well-Posedness and Exponential Ergodicity
该研究证明了含局部可积奇异漂移项的随机哈密顿系统的适定性与指数遍历性，并将结果应用于一类奇异退化McKean-Vlasov SDEs，得到其适定性与一致指数遍历性。
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.
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。
We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to $α$-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the $L^1$-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which is derived f…
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。
We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results th…
Exponential Ergodicity in $\W_1$ for SDEs with Distribution Dependent Noise and Partially Dissipative Drifts
针对McKean--Vlasov SDE，建立$L^1$-Wasserstein距离下指数遍历性的一般结果，适用于多种噪声类型，允许分布依赖系数和部分耗散漂移，显著改进现有结论。
Exponential Ergodicity for Time-Periodic McKean-Vlasov SDEs
本文研究时间周期且分布依赖的 McKean-Vlasov SDE，在耗散、部分耗散和完全非耗散三种情形下证明指数遍历性，并用颗粒介质方程说明且推广至反射情形。
We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future". The main technical result required for the analysis is a bound on the moments of the inverse of…
Exponential Ergodicity in Relative Entropy and $L^2$-Wasserstein Distance for non-equilibrium partially dissipative Kinetic SDEs
本文证明部分耗散动力学SDE在相对熵和L^2-Wasserstein距离下的指数遍历性，适用于非平衡情形，并推广至McKean-Vlasov及平均场粒子系统。
We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov's {\it apriori} estimat…
Existence, uniqueness and ergodicity for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions
本文研究了 McKean-Vlasov SDE 在分布依赖 Lyapunov 条件下的解的存在唯一性与遍历性，利用鞅表示和 Krylov-Bogolioubov 定理证明理论结果，并在特定条件下获得指数遍历性。
Exponential Ergodicity for McKean-Vlasov SDEs with Singular Interactions
针对一类具有小奇异相互作用的McKean-Vlasov随机微分方程，在1-Wasserstein和k*距离下证明了指数遍历性，并在相互作用项由小L^k范数的函数给出时，进一步得到2-Wasserstein距离和相对熵下的指数遍历性。
Let $α\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $α$-stable process in $\mathbb{R}^d$: $$ dX_t=b(X_t)dt+σ(X_{t-})d L^α_t, \quad X_0=x\in\mathbb{R}^d, $$ where $b:\mathbb{R}^d\to\mathbb{R}^d$ and $σ:\mathbb{R}^d\to\mathbb{R}^d\otimes\mathbb{R}^d$ are locally $γ$-Hölder continuous with $γ\in(0\vee(1-α)^+,1]$, $L^α_t$ is a $d$-dimensional rotationally invariant $α$-stable process. Under some dissipative and non-degenerate assumptions on $b,σ$, we show the $V$-uniformly exponential ergodicity for the semigroup $P_t$ associated with $(X_t…
Geometric ergodicity of modified Euler schemes for SDEs with super-linearity
本文提出适用于超线性漂移SDE的修正欧拉格式一般框架，利用精细基本耦合与反射耦合证明其及驯服欧拉格式在多种概率距离下的几何遍历性，并给出驯服欧拉格式不变测度与精确不变测度间的定量Wasserstein误差界。
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…
Attractivity, invariance and ergodicity for SDEs on Riemannian manifolds
针对紧连通黎曼流形上的随机微分方程，给出非线性项的一个充分条件，使得所有解的概率分布弱收敛到归一化黎曼体积测度，并应用于刻画流形上各种随机微分方程的不变测度和遍历测度。
The paper is dedicated to studying the problem of existence and uniqueness of solutions as well as existence of and exponential convergence to invariant measures for McKean-Vlasov stochastic differential equations with Markovian switching. Since the coefficients are only locally Lipschitz, we need to truncate them both in space and distribution variables simultaneously to get the global existence of solutions under the Lyapunov condition. Furthermore, if the Lyapunov condition is strengthened, we establish the exponential convergence of solutions' distributions to the unique invariant measure …
In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of th…
A class of functional differential equations are investigated. Using the Girsanov-transformation argument we establish the quadratic transportation cost inequalities for a class of finite-dimensional neutral functional stochastic differential equations and infinite-dimensional neutral functional stochastic partial differential equations under different metrics.
In this paper, we discuss hypercontractivity for the Markov semigroup $P_t$ which is generated by segment processes associated with a range of functional SDEs of neutral type. As applications, we also reveal that the semigroup $P_t$ converges exponentially to its unique invariant probability measure $μ$ in entropy, $L^2 (μ)$ and $\|\cdot\|_{\mbox{var}}$, respectively.
We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold. First, when the driving vector fields satisfy Hörmander's celebrated "Lie bracket condition," we derive explicit quantitative bounds on the inverse of the Malliavin matrix. En route to this, we provide a novel "deterministic" version of Norris's lemma for differential eq…
Irreducibility and ergodicity of SPDEs driven by pure jump noise
本文提出一种新方法，在温和且近乎必要充分的条件下，证明乘性纯跳跃噪声驱动的SPDEs/SDEs的不可约性，并应用于消除限制条件，进而获得多类奇异随机演化方程的遍历性。
We study the ergodic properties of finite-dimensional systems of SDEs driven by non-degenerate additive fractional Brownian motion with arbitrary Hurst parameter $H\in(0,1)$. A general framework is constructed to make precise the notions of ``invariant measure'' and ``stationary state'' for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solution towards the stationary one is (at least) algebraic. A lower bound on the exponent is also given.
Adam-like Algorithm with Smooth Clipping Attains Global Minima: Analysis Based on Ergodicity of Functional SDEs
本文利用泛函SDE与Markov半群遍历理论，证明带平滑裁剪的Adam型算法可收敛至正则化非凸损失的全局最小点，并给出收敛、泛化与离散化误差界。
Wellposedness, exponential ergodicity and numerical approximation of fully super-linear McKean--Vlasov SDEs and associated particle systems
研究漂移与扩散在测度和状态上具超线性增长的 McKean–Vlasov SDEs，建立适定性、混沌传播与指数遍历性，并提出强误差阶为 1/2 的粒子系统分步数值格式。
Ergodicity for regime-switching neutral stochastic functional differential equations with infinite delay
本文研究了具有无限延迟的 regime-switching 中性随机泛函微分方程的遍历性，通过分析无马尔可夫切换的中性随机泛函微分方程的well-posedness，结合耦合方法和有限分划方法，证明了有限和可数无限状态空间下的指数遍历性。
In this article, we study the ergodicity of neutral retarded stochastic functional differential equations driven by $α$-regular Volterra process. Based on the equivalence between neutral retarded stochastic functional differential equations and the stochastic evolution equation, we get the ergodicity of neutral retarded stochastic functional differential equations.
We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob--Khas'minskii theorem. The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition o…
Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients
本文研究非全局Lipschitz系数下半线性SDE不变测度的弱逼近，提出具有不变测度的线性theta投影Euler（LTPE）格式，证明其指数收敛并达到一阶弱收敛，且计算复杂度优于现有保遍历性隐式方法。
We establish the exponential convergence with respect to the $L^1$-Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equation (SDE) $$d X_t=d Z_t+b(X_t)\,d t,$$ where $(Z_t)_{t\ge0}$ is a pure jump Lévy process whose Lévy measure $ν$ fulfills $$ \inf_{x\in \R^d, |x|\le κ_0} [ν\wedge (δ_x \ast ν)]( \R^d)>0$$ for some constant $κ_0>0$, and the drift term $b$ satisfies that for any $x,y\in \R^d$, $$\langle b(x)-b(y),x-y\rangle\le \begin{cases} Φ_1(|x-y|)|x-y|,& |x-y|\le l_0; -K_2|x-y|^2,& |x-y|> l_0 \end{cases}$$ with some positive constan…
Convergence rate of nonlinear delayed neutral McKean-Vlasov SDEs driven by fractional Brownian motions
本文研究了由分数布朗运动驱动的超线性延迟中立型McKean-Vlasov随机微分方程的强收敛性，证明了解的唯一存在性、矩有界性与混沌传播，并给出了Euler-Maruyama格式的强收敛结果及数值验证。
The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with an unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of indiv…
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.
In many applications, it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite-dimensional analogue of the Langevin equation used in finite-dimensional sampling. In this paper, conditioned nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In addition, a class of preconditioned SPDEs is studied, found by applying a Green's operator to the SPDE in such a way that the invariant measure remains unchanged; such infinite dimensional evolut…
Splitting methods for stochastic Hodgkin-Huxley type systems and a localized fundamental mean-square convergence theorem
本文提出了一种适用于局部Lipschitz系数的随机微分方程局部均方收敛定理，并基于此构造了分裂方法，通过局部一致性证明和矩有界性，展示了其在Hodgkin-Huxley型系统中的优越性。
Invariant and periodic measures in classical spin systems on infinite lattices with highly degenerate noise
本文研究了无限维随机微分方程描述的经典自旋系统，在仅对一个粒子施加随机力的情况下，证明了弱马尔可夫解的存在性、唯一性和马尔可夫性质，并展示了噪声在自旋粒子中的传播，从而得到唯一自守/周期测度和几何混合性。
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