Zhao Dong, Rangrang Zhang, Tusheng Zhang
We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing property is established. This is done in the setting of kinetic solutions taking values in an L^1-weighted space.
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The Laplace transform of partial sums of the square of a non-centered Gauss-Markov process, conditioning on its starting point, is explicitly computed. The parameters of multiplicative ergodicity are deduced.
Consider the two-dimensional, incompressible Navier-Stokes equations on the torus We prove that the semigroup P_t generated by the solutions to stochastic Navier-stokes equations is asymptotically strong Feller. Moreover, we also prove that semigroup P_t is exponentially ergodic in some sense
We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. We are namely interested in a multiplicative ergodicity property used in [18] to study bifurcating processes with ancestral dependence. We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this wo…
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…
We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).
For a subfamily of multiplicative measures on integer partitions we give conditions for properly rescaled associated Young diagrams to converge in probability to a certain deterministic curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples.
We study the asymptotic behavior of solutions to stochastic evolution equations with monotone drift and multiplicative Poisson noise in the variational setting, thus covering a large class of (fully) nonlinear partial differential equations perturbed by jump noise. In particular, we provide sufficient conditions for the existence, ergodicity, and uniqueness of invariant measures. Furthermore, under mild additional assumptions, we prove that the Kolmogorov equation associated to the stochastic equation with additive noise is solvable in $L_1$ spaces with respect to an invariant measure.
Ergodicity of the viscous scalar conservation laws with a degenerate noise
本文针对环面$\boldsymbol{T}^d$上带退化噪声的粘性标量守恒律，通过引入局部化框架规避传统方法局限，在Hörmander型条件下证明其遍历性及不变测度唯一性，并刻画了不变测度有限维投影的绝对连续性。
Notes on eventual continuity and ergodicity for SPDEs
本文提出一种基于最终连续性与广义耦合技术的替代方法，用于研究乘性噪声驱动随机偏微分方程在广泛耗散系统中的渐近稳定性。
Numerical Ergodicity of Stochastic Allen--Cahn Equation driven by Multiplicative White Noise
本文建立了乘性白噪声驱动的单调SPDEs全离散格式的唯一遍历性，通过Lyapunov条件、一致矩估计及正则性推导，并验证其对随机Allen--Cahn方程任意界面厚度的适用性，数值实验支持理论结果。
In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.
Ergodicity and asymptotic limits for Langevin interacting systems with singular forces and multiplicative noises
本文研究了具有奇异力和乘法噪声的兰格vin相互作用系统，证明了经典和相对论模型的遍历性，并分析了其渐进行为及极限情况。
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…
Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise
该研究针对乘性噪声驱动的单调SPDEs数值格式，通过时间无关先验估计结合马尔可夫遍历理论证明其指数遍历性，应用于随机Allen--Cahn方程验证收敛性并解答了已有文献的相关问题。
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。
Strong and weak convergence rates for slow-fast system driven by multiplicative Lévy noises
本文研究了受α-稳定过程驱动的慢-快系统强收敛性和弱收敛率，通过耦合方法和空间周期方法推导指数遍历性，并利用热核渐进行为得到梯度估计，最终获得最优强收敛率和弱收敛率。
Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the i.i.d. supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of m…
In this paper, we established a quadratic transportation cost inequality for scalar stochastic conservation laws driven by multiplicative noise. The doubling variables method plays an important role.
Quantitative Contraction Rates for McKean-Vlasov Stochastic Differential Equations with Multiplicative Noise
本文采用耦合方法推导乘性噪声 McKean-Vlasov SDE 在 Wasserstein 距离下的显式收缩率，并据此证明时间一致的混沌传播与指数遍历性。
This article concerns the random dynamics and asymptotic analysis of the well known mathematical model, the Navier-Stokes equations. We consider the two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a \textsl{linear multiplicative white noise of Itô type} on the whole space $\mathbb{R}^2$. Firstly, we prove that the non-autonomous 2D SNSE generates a bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $\mathbb…
In this paper, we establish a small time large deviation principles for scalar stochastic conservation laws driven by multiplicative noise. The doubling of variables method plays a key role.
Convergence of the Finite Volume Method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation
本文研究了在乘性噪声下的标量守恒律的有限体积法收敛性，采用动力学公式方法证明了显式时间有限体积法的收敛性。
In this paper, we study the 2D stochastic quasi-geostrophic equation on $\mathbb{T}^2$ for general parameter $α\in(0,1)$ and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all $α\in(0,1)$. In the subcritical case $α>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions. Moreover, we prove ergodicity for the solution of the stochastic quasi-geostrophic equations in the subcritical case driven by possibly degenerate noise. The law of large numbers for the solution of the stochastic quasi-geostrophic equa…
In this work, we consider the stochastic Burgers-Huxley equation perturbed by multiplicative Gaussian noise, and discuss about the global solvability results and asymptotic behavior of solutions. We show the existence of a global strong solution of the stochastic Burgers-Huxley equation, by making use of a local monotonicity property of the linear and nonlinear operators and a stochastic generalization of localized version of the Minty-Browder technique. We then discuss about the inviscid limit of the stochastic Burgers-Huxley equation to Burgers as well as Huxley equations. By considering the…
Ergodicity of stochastic reaction-diffusion equations on unbounded domains driven by space-time white noise
研究无界域上由乘性时空白噪声驱动的随机反应扩散方程，通过设计特殊控制和比较原理，证明了不变测度的存在唯一性、指数混合性和不可约性。
Ergodicity for the randomly forced Korteweg-de Vries-Burgers equation
本文研究非可加白噪声驱动的 KdVB 方程遍历性，利用新解析性质与耦合及渐近耦合方法，证明局部化噪声下的指数遍历性和乘性噪声下的遍历性。
Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Even…
Tamed Euler-Maruyama method for SDEs with non-globally Lipschitz drift and multiplicative noise
本文针对具有超线性漂移和乘性噪声的随机微分方程，研究驯服Euler-Maruyama方法，并在部分耗散条件下证明了其在Wasserstein距离和全变差距离下的时间一致收敛速率。
Infinite ergodicity for geometric Brownian motion
本文研究几何布朗运动及其推广的渐近分布特性，通过无限遍历性方法探讨其在不同离散化参数下的概率分布行为。
In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.
Ergodicity and Mixing of invariant capacities and applications
该研究引入公共条件期望与骨架概率方法，研究不变上概率的遍历定理与弱混合性，给出多个应用与实例，并推广至经典遍历理论与非不变概率场景。
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。
Ergodicity for the Dean--Kawasaki Equation with Dirichlet Boundary Conditions: Taming the Square-Root
本文研究了具有平方根型噪声的德恩-卡瓦基方程在狄利克雷边界条件下的遍历性，证明了其在不同噪声条件下收敛速率的差异。
We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniqueness and continuous dependence, in the form of pathwise $L^1$-contraction, as well as some explicit estimates. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced…
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for $V : \mathbb{R}^d \to \mathbb{R}$ a potential function to minimize, we consider the stochastic equation $dY_t = - σσ^\top \nabla V(Y_t) dt + a(t)σ(Y_t)dW_t + a(t)^2Υ(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $σ: \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $Υ$ is a correction term. This setting can be applied to optimization problems arising in Machine Learn…
Ergodicity for stochastic T-monotone parabolic obstacle problems
本文研究了由Sobolev空间上T-单调算子控制、受乘性噪声驱动的障碍问题，证明了其解定义了Markov-Feller半群，并在适当假设下确立了遍历不变测度的存在性与唯一性。
We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.
We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $\R^N$ with quasilinear multiplicative ''rough path'' dependence by considering inhomogeneous fluxes and a single rough path like, for example, a Brownian motion. Following our previous note where we considered spatially independent fluxes, we introduce the notion of pathwise stochastic entropy solutions and prove that it is well posed, that is we establish existence, uniqueness and continuous dependence in the form of a (pathwise) $L^1$-contraction. Our approach is motivated by t…
We study a BGK-like approximation to hyperbolic conservation laws forced by a multiplicative noise. First, we make use of the stochastic characteristics method and establish the existence of a solution for any fi xed parameter $\varepsilon$. In the next step, we investigate the limit as $\varepsilon$ tends to 0 and show the convergence to the kinetic solution of the limit problem.
We study the asymptotic properties of the stochastic Cahn-Hilliard equation with the logarithmic free energy by establishing different dimension-free Harnack inequalities according to various kinds of noises. The main characteristics of this equation are the singularities of the logarithmic free energy at 1 and --1 and the conservation of the mass of the solution in its spatial variable. Both the space-time colored noise and the space-time white noise are considered. For the highly degenerate space-time colored noise, the asymptotic log-Harnack inequality is established under the so-called ess…
Stochastic Fractional Conservation Laws: Large deviation principle, Central limit theorem and Moderate deviation principle
本文利用弱收敛与加倍变量方法，在动力学框架下证明带小乘性噪声的随机分数阶守恒律的 Freidlin-Wentzell 型大偏差原理、中心极限定理及中偏差原理。
We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and Souganidis who considered spatially independent and inhomogeneous fluxes with multiple paths and a single driving singular path respectively. The approach is motivated by the theory of stochastic viscosity solutions which relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the ki…
Strong Feller property, irreducibility, and uniqueness of the invariant measure for stochastic PDEs with degenerate multiplicative noise
本文研究了退化乘性噪声下随机PDEs的强Feller性质和不可约性，并证明了在无强耗散假设下不变测度的唯一性，通过测量解向势垒的积累，赋予退化噪声系数的逆意义。
In this paper, we are concerned with a operator splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Levy noise. More specifically, using a variant of classical Kruzkov's doubling of variable approach, we show that the approximate solutions generated by the splitting scheme converges to the unique stochastic entropy solution of the underlying problems.Finally, the convergence analysis is illustrated by several numerical examples.
Hypocoercivity for infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise
本文研究无穷维非线性退化随机微分方程，建立生成算子的耗散性并构造解过程，应用次协调性方法证明指数收敛，推广至反应-扩散及Cahn-Hilliard型方程。
Well-posedness and invariant measure for quasilinear parabolic SPDE on a bounded domain
本文研究了有界区域上带一般乘性噪声的拟线性抛物型随机偏微分方程，在 $L^{1}$ 框架下证明了解的适定性、比较结果与收缩性质，并在特定扩散与噪声条件下证明了不变测度的存在性、唯一性与遍历性。
In this work, we consider the two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations and examine some asymptotic behaviors of its strong solution. We establish the asymptotic log-Harnack inequality for the transition semigroup associated with the SCBF equations driven by additive as well as multiplicative degenerate noise via the asymptotic coupling method. As applications of the asymptotic log-Harnack inequality, we derive the gradient estimate, asymptotic irreducibility, asymptotic strong Feller property, asymptotic heat kernel estimate and ergodicity. Whenever…
From Stochastic Zakharov System to Multiplicative Stochastic Nonlinear Schr{ö}dinger Equation
研究时间白噪声驱动、空间有色噪声的 Zakharov 系统在离子声速趋于无穷时向乘性随机非线性 Schr{ö}dinger 方程的收敛性，通过扩散近似框架与扰动测试函数方法克服噪声导致的奇异性，并证明一维情形下解的存在唯一性与依概率收敛。
Ergodicity of the Anderson $Φ_2^4$ model
本文研究奇异空间白噪声环境中 $Φ_2^4$ 随机量子化方程的长时间行为，证明全局解、强 Feller 性及指数收敛到唯一不变测度，并将方法推广到非平移不变及向量值情形。
Large time behaviour for a class of 2D and 3D stochastic non-Newtonian fluids of differential types: Attractors and invariant measures
本文研究了随机三级流体方程在一般无界域上的大时间行为，证明了随机吸引子和不变测度的存在性，解决了随机吸引子、不变测度和遍历性方面的多个开放问题。
We establish new conditions for obtaining uniform bounds on the moments of discrete-time stochastic processes. Our results require a weak negative drift criterion along with a state-dependent restriction on the sizes of the one-step jumps of the processes. The state-dependent feature of the results make them suitable for a large class of multiplicative-noise processes. Under the additional assumption of Markovian property, new result on ergodicity has also been proved. There are several applications to iterative systems, control systems, and other dynamical systems with state-dependent multipl…
Well-posedness and large deviations for the obstacle problem of first-order stochastic conservation laws
该研究针对乘性噪声驱动的一阶标量守恒律障碍问题，适配障碍替换策略到动力学公式，消除障碍-噪声相容性条件，建立适定性并证明$L^1$空间的Freidlin-Wentzell大偏差原理，为抛物型反射SPDE的大偏差理论提供双曲型对应结果。
In this paper we study a stochastic version of an inviscid shell model of turbulence with multiplicative noise. The deterministic counterpart of this model is quite general and includes inviscid GOY and Sabra shell models of turbulence. We prove global weak existence and uniqueness of solutions for any finite energy initial condition. Moreover energy dissipation of the system is proved in spite of its formal energy conservation.
Gaussian fluctuations for the stochastic wave equation with drift
本文研究带非线性漂移和乘性高斯噪声的随机波动方程空间平均的高斯波动，建立定量中心极限定理，误差界分别为可积情形下的R^{-d/2}和Riesz核下的R^{-\eta/2}，并获泛函中心极限定理。
A stochastic version of an inviscid dyadic model of turbulence, with multiplicative noise, is proved to exhibit energy dissipation in spite of the formal energy conservation. As a consequence, global regular solutions cannot exist. After some reductions, the main tool is the escape bahavior at infinity of a certain birth and death process.
The essential m-dissipativity for degenerate infinite dimensional stochastic Hamiltonian systems and applications
本文在$L^2(μ^Φ)$上建立了带乘性噪声的退化无限维随机哈密顿系统对应Kolmogorov算子的基本m-耗散性，并利用亚强制性方法构造$μ^Φ$-不变Hunt过程，进而证明指数遍历性，最终应用于退化随机反应扩散方程。
An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes
本文提出了一种包含跳跃扩散噪声的随机Allen-Cahn方程，用于建模生物损伤与修复过程，并分析了其数学性质与数值模拟方法。
Invariant measures of randomized quantum trajectories
本文研究随机化探针可观测量的量子轨迹性质，证明非奇异随机化使轨迹纯化并具有唯一不变测度，引入乘法原生性概念并计算典型量子通道的不变测度。
Solutions to the stochastic thin-film equation for initial values with non-full support
本文研究一维环面上带乘性Stratonovich噪声的随机薄膜方程，通过基于α-熵耗散和质量守恒的紧性论证，证明非负初值下鞅解存在，在迁移率、初值类型和噪声正则性三方面推进了现有结果。
The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener-processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The pr…
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