Giuseppe Da Prato, Boris L. Rozovskii, Michael Röckner, Feng-Yu Wang
Explicit conditions are presented for the existence, uniqueness and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.
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The long time behaviour of solutions to stochastic porous media equations on smooth bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-estimates the existence and uniqueness of invariant measures with optimal bounds on the rate of mixing are proved. Along the way the existence and uniqueness of entropy solutions is shown.
The long time behaviour of solutions to generalised stochastic porous media equations on bounded domains with Dirichlet boundary data is studied. We focus on a degenerate form of nonlinearity arising in self-organised criticality. Based on the so-called lower-bound method, the existence and uniqueness of an invariant measure is proved.
The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.
One proves that the stochastic porous media equation in 3-D has a unique nonnegative solution for nonnegative initial data in $H^{-1}(\mathcal O)$ if the nonlinearity is monotone and has polynomial growth.
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.
McKean-Vlasov SPDEs with coefficients exhibiting locally weak monotonicity: existence, uniqueness, ergodicity, exponential mixing and limit theorems
本文针对具有局部弱单调性系数的McKean-Vlasov SPDE，利用Galerkin投影和广义耦合方法证明了解的存在唯一性，进而建立指数遍历性、指数混合性以及强大数定律和中心极限定理。
Existence, uniqueness and ergodicity for the centered Fleming-Viot process
本文通过鞅问题刻画中心化 Fleming–Viot 过程，利用渐近展开与弱化对偶方法证明有限矩初始条件下解的存在唯一性，并确立其全变差指数遍历性及不变测度。
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。
We study the ergodic theory of stationary directed nearest-neighbor polymer models on $\mathbb Z^2$, with i.i.d. weights. Such models are equivalent to specifying a stationary distribution on the space of weights and correctors that satisfy certain consistency conditions. We show that for prescribed weight distribution and corrector mean, there is at most one stationary polymer distribution which is ergodic under the $e_1$ or $e_2$ shift. Further, if the weights have more than two moments and the corrector mean vector is an extreme point of the superdifferential of the limiting free energy, th…
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 prove the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly \cite{Ha-Ma}, we prove the uniqueness of invariant measures for the corresponding transi…
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.
This work focuses on a class of regime-switching jump diffusion processes, in which the switching component has countably infinite many states or regimes. The existence and uniqueness of the underlying process are obtained by an interlacing procedure. Then the Feller and strong Feller properties of such processes are derived by the coupling method and an appropriate Radon-Nikodym derivative. Finally the paper studies exponential ergodicity of regime-switching jump-diffusion processes.
One proves existence and uniqueness of strong solutions to stochastic porous media equations under minimal monotonicity conditions on the nonlinearity. In particular, we do not assume continuity of the drift or any growth condition at infinity.
Ergodicity of Stochastic two-phase Stefan problem driven by pure jump Lévy noise
本文研究由纯跳Lévy噪声驱动的随机两相Stefan问题，证明其强解存在唯一并建立遍历性，同时刻画不变测度支撑集以给出平稳解的正则性。
We study a model of the motion by mean curvature of an (1+1) dimensional interface in a 2D Brownian velocity field. For the well-posedness of the model we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution equations in the variational framework of Krylov Rozovskii, replacing the standard coercivity assumption by a Lyapunov type condition. Ergodicity is established for the case of additive noise, using the lower bound technique for Markov semigroups by Komorowski, Peszat and Szarek
In this paper, we establish the existence, uniqueness and attraction properties of an invariant measure for the real Ginzburg-Landau equation in the presence of a degenerate stochastic forcing acting only in four directions. The main challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. To achieve this, we propose a condition which only requires four noises
Using the generalized variational framework, the strong/weak existence and uniqueness of solutions are derived for a class of distribution dependent stochastic porous media equations on general measure spaces, which also extends the classical well-posedness result of quasilinear SPDE to the distribution dependent case.
Based on a new explicit representation of the solution to the Poisson equation with respect to single birth processes, the unified treatment for various criteria on classical problems (including uniqueness, recurrence, ergodicity, exponential ergodicity, strong ergodicity, as well as extinction probability etc.) for the processes are presented.
We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we study statistically steady states of the system and find that they possess regularities compatible with those of the weak solutions. Moreover, if sufficiently many directions in the phase space are stochastically forced, we employ the \emph{generalized coupling} approach to establish the existence and uniqueness of the invariant probability measure to which the system is exponentially attractive. This extends ergodicity…
We consider a stochastic electroconvection model describing the nonlinear evolution of a surface charge density in a two-dimensional fluid with additive stochastic forcing. We prove the existence and uniqueness of solutions, we define the corresponding Markov semigroup, and we study its Feller properties. When the noise forces enough modes in phase space, we obtain the uniqueness of the smooth invariant measure for the Markov transition kernels associated with the model.
This paper considers multidimensional jump type stochastic differential equations with super linear growth and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient non-Lipschitz conditions for pathwise uniqueness. The non confluence property for solutions is investigated. Feller and strong Feller properties under non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic di…
We consider a stochastic version of a system of coupled two equations formulated by Burgers with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness of the solution as well as the irreducibility property of such system were given by Twardowska and Zabczyk. In the paper the existence of a unique invariant measure is investigated. The paper generalizes the results of Da Prato, Debussche and Temam, and Da Prato and Gatarek, dealing with one equation describing the turbulent motion only.
We establish the existence and uniqueness of an ergodic invariant measure for 2D fractionally dissipated stochastic Euler equations on the periodic box, for any power of the dissipation term.
The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations $ω$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.
The existence of stationary distributions to distribution dependent stochastic differential equations are investigated by using the ergodicity of the associated decoupled equation and the Schauder fixed point theorem. By using Zvonkin's transformation, we also establish the existence result for equations with singular coefficients. Instead of the uniqueness, the non-uniqueness of stationary distributions are considered for equations with regular coefficients. Concrete examples including McKean-Vlasov stochastic equations with the quadratic interaction and the non-quadratic interaction, and equ…
We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering the space variable-independent solutions only. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and we obtain also results on attractivity towards an invariant measure. We also present a suitable numerical scheme for approximating the solutions subject to a sphere constraint.
We consider a porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients.
Stochastic Hamiltonian Type Jump Diffusion Systems with Countable Regimes: Strong Feller Property and Exponential Ergodicity
本文研究切换分量具有可数无限状态的随机哈密顿型跳扩散系统，证明了解的存在唯一性、Feller性与强Feller性，并在Foster-Lyapunov漂移条件下建立了指数遍历性。
Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise
本文针对高度退化的纯跳Lévy噪声驱动的二维Navier-Stokes方程，利用Malliavin分析与预想随机分析建立e-property和弱不可约性，从而证明遍历性并解决长期未决问题。
We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for $σ$-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called $N$-functions in the theory of Orlicz spaces.
This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has a countably infinite state space. After establishing the existence and uniqueness of a global weak solution via the martingale approach under very mild conditions, the paper next proves the strong Feller property for regime-switching stochastic damping Hamiltonian systems by the killing technique together with some resolvent and transition probability identities. The commonly used continuity assumption for the switching rates $q_{kl}(\cdot)$ in the literature i…
Ergodicity for eventually continuous Markov--Feller semigroups on Polish spaces
本文研究波兰空间上马尔可夫--费勒半群在极弱正则性条件下的遍历性，证明了Cesàro平均的弱收敛性，刻画了Cesàro最终连续性、e-性质与弱-*平均遍历性的关系，并给出了不变测度存在唯一性准则及遍历分解性质，最后应用于多个非平凡系统。
We prove the existence and uniqueness of non-negative entropy solutions of the obstacle problem for stochastic porous media equations. The core of the method is to combine the entropy formulation with the penalization method.
Ergodicity for stochastic 2D Boussinesq equations with a highly degenerate pure jump Levy noise
研究仅作用于温度方程的退化纯跳跃Lévy噪声驱动的随机二维Boussinesq方程的遍历性，通过Malliavin分析和随机分析证明不变测度的存在唯一性。
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。
We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbe…
Ergodicity for a class of stochastic Hasegawa-Mima equations with dissipation
研究有界域中加性噪声扰动的随机耗散Hasegawa-Mima方程的遍历性，利用渐近耦合技术得到不变测度的唯一性。
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.
We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.
The paper analyses a model in surface growth, where uniqueness of weak solutions seems to be out of reach. We provide the existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under non-degeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure.
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 prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E, \mathscr{B}(E), μ)$, and the Laplacian replaced by a self-adjoint operator $L$. In the case of Lipschitz nonlinearities $Ψ$, we in particular generalize previous results for open $E\subset \mathbb{R}^d$ and $L\!\!=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed i…
We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the $2$-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author to a situation where the invariant measure is not explicitly known.
In this paper, we first show the well-posedness of the SDEs driven by Lévy noises under mild conditions. Then, we consider the existence and uniqueness of periodic solutions of the SDEs. To establish the ergodicity and uniqueness of periodic solutions, we investigate the strong Feller property and the irreducibility of the corresponding time-inhomogeneous semigroups when both small and large jumps are allowed in the equations. Some examples are presented to illustrate our results.
Ergodicity for SPDEs driven by divergence-free transport noise
本文研究了由分歧自由传输噪声驱动的SPDEs的遍历性，证明强混合噪声可确保唯一不变测度，即使确定性部分存在多个稳态。
Existence and uniqueness of solutions to the stochastic porous media equation $dX-\Dψ(X) dt=XdW$ in $\rr^d$ are studied. Here, $W$ is a Wiener process, $ψ$ is a maximal monotone graph in $\rr\times\rr$ such that $ψ(r)\le C|r|^m$, $\ff r\in\rr$, $W$ is a coloured Wiener process. In this general case the dimension is restricted to $d\ge 3$, the main reason being the absence of a convenient multiplier result in the space $\calh=\{φ\in\mathcal{S}'(\rr^d);\ |ξ|(\calfφ)(ξ)\in L^2(\rr^d)\}$, for $d\le2$. When $ψ$ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sob…
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。
Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations
该研究利用拟平稳与拟遍历测度框架，证明了加性柱噪声扰动的反应扩散系统相关测度的存在唯一性及收敛性，基于谱隙条件刻画了系统在确定性系统不变流形邻域的条件行为，且方法不限于小噪声情形。
Ergodicity results for the open KPZ equation
本文为开放边界KPZ方程的不变测度提供了存在性新证明和唯一性双证明，结合强Feller性质得到全变差指数收敛，并构造紧致状态空间以扩展适用范围。
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…
Consider a $1$-dimensional centered Gaussian process $W$ with $α$-Hölder continuous paths on the compact intervals of $\mathbb R_+$ ($α\in ]0,1[$) and $W_0 = 0$, and $X$ the local solution in rough paths sense of Jacobi's equation driven by the signal $W$. The global existence and the uniqueness of the solution are proved via a change of variable taking into account the singularities of the vector field, because it doesn't satisfy the non-explosion condition. The regularity of the associated Itô map is studied. By using these deterministic results, Jacobi's equation is studied on probabilistic…
We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analysed in detail.
We study the surface quasi-geostrophic equation with an irregular spatial perturbation $$ \partial_{t }θ+ u\cdot\nablaθ= -ν(-Δ)^{γ/2}θ+ ζ,\qquad u=\nabla^{\perp}(-Δ)^{-1}θ, $$ on $[0,\infty)\times\mathbb{T}^{2}$, with $ν\geq 0$, $γ\in [0,3/2)$ and $ζ\in B^{-2+κ}_{\infty,\infty}(\mathbb{T}^{2})$ for some $κ>0$. This covers the case of $ζ= (-Δ)^{α/2}ξ$ for $α<1$ and $ξ$ a spatial white noise on $\mathbb{T}^{2}$. Depending on the relation between $γ$ and $α$, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures \cite{Hai14}. Based on purely analy…
Ergodicity for the fractional Magneto-Hydrodynamic equations driven by a degenerate pure jump noise
本文研究了由高度退化纯跳跃 Lévy 噪声驱动的随机 2D 分数阶磁流体动力学方程的遍历性，针对噪声仅作用于四个方向的高退化情形，结合 Malliavin 变分法证明半群 e-property 与弱不可约性，确立了不变测度的唯一性。
The ergodicity of nonlinear McKean-Vlasov stochastic differential equations with common noise
研究具有公共噪声的非线性麦克凯恩 - 弗拉索夫随机微分方程的遍历性，通过提升系统构造半群，利用耦合方法证明不变测度相关结论，经解耦导出大数定律，建立混沌传播及收敛性，并用示例验证。
The use of higher-order stochastic processes such as nonlinear Markov chains or vertex-reinforced random walks is significantly growing in recent years as they are much better at modeling high dimensional data and nonlinear dynamics in numerous application settings. In many cases of practical interest, these processes are identified with a stochastic tensor and their stationary distribution is a tensor $Z$-eigenvector. However, fundamental questions such as the convergence of the process towards a limiting distribution and the uniqueness of such a limit are still not well understood and are th…
Consider the non-linear stochastic fractional-diffusion equation \begin{eqnarray*} \left \{\begin{array}{lll} \frac{\partial}{\partial t}u(x,t)= -( Δ)^{α/2} u^m(x,t) + σ(u(x,t)) \dot{W}(x,t),\, x\in \mathbb{R}^d,t>0, u(x,0)= u_0(x),\,\,\, x\in\mathbb{R}^d \end{array}\right. \end{eqnarray*} with initial data $u_0(x)$ an $L^1(\mathbb{R}^d)$ function, $0<α<2$, and $m>0$. There is no mild solution defined for the above equation because its corresponding heat kernel representation does not exist. We attempt to make sense of the above equation by establishing the existence and uniqueness result via …
We give sufficient conditions for existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Lévy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.
On the Long-time Dynamics and Ergodicity of the Stochastic Nernst-Planck-Navier-Stokes System
本文研究随机Nernst-Planck-Navier-Stokes电扩散系统的长时间动力学与遍历性，建立了全局路径解的存在唯一性、Feller性质、不变遍历测度的存在性与唯一性，并在两种等扩散离子情形下证明了无条件唯一指数遍历性。
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