Roland Schnaubelt, Mark Veraar
We study the wellposedness and pathwise regularity of semilinear non-autonomous parabolic evolution equations with boundary and interior noise in an $L^p$ setting. We obtain existence and uniqueness of mild and weak solutions. The boundary noise term is reformulated as a perturbation of a stochastic evolution equation with values in extrapolation spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space-time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).
We establish a new version of the stochastic Strichartz estimate for the stochastic convolution driven by jump noise which we apply to the stochastic nonlinear Schrödinger equation with nonlinear multiplicative jump noise in the Marcus canonical form. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to stochastic nonlinear Schrödinger equation in $L^2(\RR^n)$ with either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.
A Random Change of Variables and Applications to the Stochastic Porous Medium Equation with Multiplicative Time Noise
引入变量变换，将一类带乘性噪声的非线性随机发展方程化为相应的确定性方程。最后，将所得结果应用于随机多孔介质方程。
The primitive equations with stochastic wind driven boundary conditions
本文研究随机风驱动边界条件下地球物理流原始方程，提出结合随机与确定性方法的严格处理，证明在各向异性临界空间中存在唯一局部路径wise解。
In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction-diffusion equations, stochastic fractional Burgers equation, stoch…
An impact of stochastic dynamic boundary conditions on the evolution of the Cahn-Hilliard system
非线性系统经常受到随机因素的影响。有时噪声通过物理边界进入系统，这导致随机动态边界条件。与静态边界条件相反，动态边界条件涉及边界上系统状态变量的时间导数和空间导数。虽然随机\emph{static}（Neumann或Dirichet型）边界条件已被应用于随机偏微分方程，但对随机\emph{dynamic}边界条件的动力学影响知之甚少。本文研究了材料科学中的Cahn-Hilliard方程在随机动态边界条件下的长期动力学行为。我们发现，随机吸引子的维数估计增加的系数的动态项的随机动态边界条件的减少。然而，随机吸引子的维数不受相应的随机静态边界条件的影响。
This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinear wave equation with a random dynamical boundary condition. The white noises are taken into account not only in the model equation defined on a domain perforated with small holes, but also in the dynamical boundary condition on the boundaries of the small holes. An effective homogenized, macroscopic model is derived in the sense of probability distribution, which is a new stochastic wave equation on a unified domain, without small holes, with a usual static boundary condition.
In this paper, we investigate a stochastic Hardy-Littlewood-Sobolev inequality. Due to the stochastic nature of the inequality, the relation between the exponents of intgrability is modified. This modification can be understood as a regularization by noise phenomenon. As a direct application, we derive Strichartz estimates for the white noise dispersion which enables us to address a conjecture from [3].
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $α>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $α$-Hölder continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $C^1$ domains are proved to be $α$-Hölder continuous up to the boundary with some $α>0$.
In this article, we study Stochastic mass critical nonlinear Schrödinger equations with a multiplicative noise in 3D with a slight time decay ($\langle t \rangle^{-ε}$), and prove associated space-time bound and scattering behavior.
Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula.
The fluctuation behaviour of the stochastic point vortex model with common noise
本文研究带公共噪声的随机点涡模型的涨落行为，采用鞅方法结合局部化论证证明涨落过程依分布收敛于线性随机演化方程的唯一概率强解，并建立该模型到条件McKean-Vlasov方程的强收敛性。
We prove that the any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e.all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.
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 prove the Freidlin-Wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave (NLW) equation with white noise. We do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise. This allows to provide the first such result in the PDE setting.
In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schrödinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.
We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes are forced) via Kolmogorov equation approach.
Exact variations for stochastic heat equations driven by space--time white noise
本文计算了乘性时空白噪声驱动下一维随机热方程解的精确空间二次变分和时间四次变分。
On the local linearization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing
针对乘性时空白噪声驱动的一维随机波动方程，研究建立了其双参数线性局部化结论，为该类方程的局部性质分析提供支撑。
We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations with a polynomially growing quasi-monotone nonlinearity and multiplicative Poisson noise. We also study existence and uniqueness of invariant measures for the associated semigroup in the Markovian case. A key role is played by a new maximal inequality for stochastic convolutions in $L_p$ spaces.
We study the the sharp interface limit of $\varepsilon$-dependent two dimensional stochastic Cahn-Hilliard equation driven by space-time white noise and conservative noise as $\varepsilon\to 0$. In the case when the noise is sufficiently small, by comparing the solutions to equation (1.1) with the approximation solution constructed in [ABC94], we show that the limit of the solutions is also solutions to the deterministic Hele-Shaw problem.
This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting skill is used to derive the approximating equation of the system in the sense of probability distribution, when the singular perturbation parameter is sufficiently small. The approximating equation is a stochastic parabolic equation when the power exponent of singular perturbation parameter is in $[1/2, 1)$, but a deterministic hyperbolic (wave) equation when the power exponent is in $(1, +\infty)$.
Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions
本文研究二氧化硫与碳酸钙石材化学反应驱动的半直线非线性反应扩散系统，边界条件为Jacobi过程，通过分裂策略证明了温和解的全局存在性与轨道唯一性。
We study the stochastic Camassa-Holm equation with pure jump noise. We prove that if the initial condition of the solution is a solitary wave solution of the unperturbed equation, the solution decomposes into the sum of a randomly modulated solitary wave and a small remainder. Moreover, we derive the equations for the modulation parameters and show that the remainder converges to the solution of a stochastic linear equation as amplitude of the jump noise tends to zero.
We study a class of stochastic evolution equations with a dissipative forcing nonlinearity and additive noise. The noise is assumed to satisfy rather general assumptions about the form of the covariance function; our framework covers examples of Gaussian processes, like fractional and bifractional Brownian motion and also non Gaussian examples like the Hermite process. We give an application of our results to the study of the stochastic version of a common model of potential spread in a dendritic tree. Our investigation is specially motivated by possibility to introduce long-range dependence i…
In this paper we focus on the stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise. We first establish two new cancellation properties on pseudo-differential operators, which play a key role in energy estimate. Then, we obtain results on local solution, blow-up criterion and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data are also studied for the multiplicative noise case.
Stochastic Two-Dimensional Anisotropic Magnetohydrodynamics System with Multiplicative Noise
本文研究速度耗散仅水平方向的随机二维各向异性磁流体系统，在乘性噪声下证明了全局鞅解和概率弱解的存在性及路径唯一性。
In this article we investigate the spatial Sobolev regularity of mild solutions to stochastic Burgers equations with additive trace class noise. Our findings are based on a combination of suitable bootstrap-type arguments and a detailed analysis of the nonlinearity in the equation.
In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.
The inviscid 2D Boussinesq system with thermal diffusivity and multiplicative noise of transport type is studied in the $L^2$-setting. It is shown that, under a suitable scaling of the noise, weak solutions to the stochastic 2D Boussinesq equations converge weakly to the unique solution of the deterministic viscous Boussinesq system. Consequently, the transport noise asymptotically regularizes the inviscid 2D Boussinesq system and enhances dissipation in the limit.
We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path $L^{\infty}$ bounds for the second derivative of solutions to such PDE are shown. These bounds are expressed as solutions to reflected SDE and are shown to be optimal.
This paper is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential integral equations with nonlinear Neumann boundary condition. Using the recently developed theory on generalized backward doubly stochastic differential equations driven by a Lévy process, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula.
Under non-periodic boundary conditions, we consider the long-time behavior for stochastic 2D nematic liquid crystals flows with velocity and orientations perturbed by additive noise and multiplicative noise respectively. It is the first result for the long-time behavior of stochastic nematic liquid crystals under Dirichlet boundary condition for velocity field and Neumann boundary condition for orientation field.
In this paper, we prove a particle approximation, in the sense of the propagation of chaos, of a Lagrangian stochastic model submitted to specular boundary condition and satisfying the mean no-permeability condition.
Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions
为了理解物理边界上的随机影响对多尺度系统演化的影响，研究了一个快速随机动力边界条件下的随机偏微分方程模型.模型中的噪声和边界条件中的噪声都是加性的。通过将随机\emph{dynamical}边界条件简化为一个简单的边界条件，导出了一个有效的方程。有效系统仍然是一个随机偏微分方程。通过建立正态偏差和大偏差原理，给出了原随机系统解与有效解的定量比较。也就是说，正态偏差的渐近特征，而大偏差的速率和速度的估计。
Optimal control of stochastic differential equations with dynamical boundary conditions
本文研究了一类具有非标准边界动力和控制的随机柯西发展问题的最优控制问题。该模型由一个无限维动力系统和一个有限维动力系统耦合而成，后者描述了内部系统的边界条件。换句话说，我们关注的是非标准边界条件，因为边界处的值由不同的随机微分方程决定。
In this paper we propose and analyze a class of $N$-player stochastic games that include finite fuel stochastic games as a special case. We first derive sufficient conditions for the Nash equilibrium (NE) in the form of a verification theorem. The associated Quasi-Variational-Inequalities include an essential game component regarding the interactions among players, which may be interpreted as the analytical representation of the conditional optimality for NEs. The derivation of NEs involves solving first a multi-dimensional free boundary problem and then a Skorokhod problem. We call it a "movi…
The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction-diffusion equations, the stochastic $p$-Laplace equation and stochastic porous media equations. Besides classical Brownian motion, we also include space-time fractional Brownian Motion and space-time Lévy noise as admissible random perturbations. Moreover, cases where the attractor consists of a single point are considered and bounds for the speed of attraction…
Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions
考虑了随机影响下的微观非均匀系统。随机性在小尺度障碍物的物理边界以及物理介质的内部进入系统。这个系统是由一个随机的偏微分方程定义的域上穿孔的小孔（障碍物或异质性），连同随机动态边界条件的边界上的这些小孔。导出了这种微观非均匀随机系统的均匀化宏观模型。该均匀化有效模型是一个新的随机偏微分方程，它定义在一个统一的无小孔区域上，仅具有静态边界条件。实际上，随机动力边界条件被均匀化，但随机力对小孔边界的影响被量化为均匀化随机偏微分方程中的额外随机项。此外，均匀化模型的有效性是合理的，通过显示的解决方案的微观模型收敛到有效的宏观模型的概率分布，作为小孔的大小减小到零。
Linear stochastic differential equations with functional boundary conditions
考虑[0，1]上的线性n阶随机微分方程，其线性边界条件由[0，1]的有限子集支持.我们研究了这类问题解的一些特征，特别是它的马尔可夫型条件独立性。
In this article, we study the stability of solutions to 3D stochastic primitive equations driven by fractional noise. Since the fractional Brownian motion is essentially different from Brownian motion, lots of stochastic analysis tools are not available to study the exponential stability for the stochastic systems. Therefore, apart from the standard method for the case of Brownian motion, we develop a new method to show that 3D stochastic primitive equations driven by fractional noise converge almost surely exponentially to the stationary solutions. This method may be applied to other stochast…
Stochastic evolution equations with rough boundary noise
本文结合受控粗糙路径与外推算子理论，证明乘性Neumann边界噪声（如分数布朗运动）下随机发展方程解的全局存在性与流，并在Young框架下推广至Dirichlet边界噪声。
We investigate a class of stochastic integro differential equations driven by Levy noise.
In this paper, we introduce the idea of stochastic integrals with respect to an increasing process in the $G$-framework and extend $G$-Itô's formula. Moreover, we study the solvability of the scalar valued stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions (RGSDEs).
In this paper we will show that the solution of 1D stochastic parabolic equation with additive noise converges to a martingale (independent upon space variable) when we rescale noise at the extremum points of the process.
Maximal regularity for stochastic convolutions driven by Levy noise
我们证明了Da Prato和Lunardi的结果对于Lévy过程驱动的随机卷积是有效的。
Stochastic heat flow is a black noise
本文证明随机热流是黑噪声，并推论由平滑时空白噪声驱动的2d随机热方程在标度极限下渐近独立于该噪声。
In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal $L^2$-regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms…
We prove the exponential stability of the zero solution of a stochastic differential equation with a Hölder noise, under the strong dissipativity assumption. As a result, we also prove that there exists a random pullback attractor for a stochastic system under a multiplicative fractional Brownian noise.
We consider a reaction-diffusion equation on a network subjected to dynamic boundary conditions, with time delayed behaviour, also allowing for multiplicative Gaussian noise perturbations. Exploiting semigroup theory, we rewrite the aforementioned stochastic problem as an abstract stochastic partial differential equation taking values in a suitable product Hilbert space, for which we prove the existence and uniqueness of a mild solution. Eventually, a stochastic optimal control application is studied.
The flow of nematic liquid crystals can be described by a highly nonlinear stochastic hydrodynamical model, thus is often influenced by random fluctuations, such as uncertainty in specifying initial conditions and boundary conditions. In this article, we consider the $2$-D stochastic nematic liquid crystals with the velocity field perturbed by affine-linear multiplicative white noise, with random initial data and random boundary conditions. Our main objective is to establish the global well-posedness of the stochastic equations under certain sufficient Malliavin regularity of the initial condi…
A stochastic affine evolution equation with bilinear noise term is studied where the driving process is a real-valued fractional Brownian motion. Stochastic integration is understood in the Skorokhod sense. Existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained
Burkholder-Davis-Gundy type Inequalities of the Itô stochastic integral with respect to Levy noise on Banach spaces
给出了Banach值Levy噪声下伊藤积分的Burkholder-Davis-Gundy型不等式.
Fang-Wu\cite{FW17} presented a explicit spectral gap for the O-U process on path space over a Riemannian manifold without boundary under the bounded Ricci curvature conditions. In this paper, we will extend these results to the case of the Riemannian manifold with boundary. Moreover, we also derive the similar results for the stochastic heat process.
We describe sufficient conditions on the reaction terms and multiplicative noise terms of a stochastic reaction-diffusion equation that guarantee that the solutions never explode. Both the reaction term and multiplicative noise terms are allowed to grow superlinearly.
We consider a controlled state equation of parabolic type on the halfline $(0,+\infty)$ with boundary conditions of Dirichlet type in which the unknown is equal to the sum of the control and of a white noise in time. We study finite horizon and infinite horizon optimal control problem related by menas of backward stochastic differential equations.
Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case
本文证明了当系数是指数为$γ>3/4$的Hölder连续时，带乘性白噪声的抛物型随机偏微分方程解的路径唯一性.证明的方法是一个无限维版本的山田渡边参数普通随机微分方程。
In this article we consider multiplicative operator-valued white noise functionals related to a stochastic flow. A generalization of the Krylov-Veretennikov expansion is presented. An analog of such expansion for the Arratia flow is derived.
Semigroups on generalized Sobolev spaces associated with Laplacians with applications Stochastic PDEs with singular boundary conditions
该研究考察带奇异边界条件区域对应的拉普拉斯算子及其生成的广义Sobolev空间半群，并给出其在带奇异边界条件的随机偏微分方程中的应用。
We study stochastic SQG equations on the torus $\mathbb{T}^2$ with multiplicative transport noise in the $L^2$-setting. Under a suitable scaling of the noise, we show that the solutions converge weakly to the unique solution to the deterministic dissipative SQG equation. A similar scaling limit result is proved also for the stochastic 2D inviscid critical Boussinesq equations.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。