Invariant measures for stochastic heat equations with unbounded coefficients

Assing, Sigurd; Manthey, Ralf

doi:10.1016/s0304-4149(02)00211-9

Cited by 21 publications

(36 citation statements)

References 7 publications

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“…In paper [27], the invariant measure of stochastic heat equation with coefficients unbounded was considered via comparison method. Paper [28] proved a large deviation principle for the invariant measures of a class of reaction diffusion equations with multiplicative noise and non-Lipschitz reaction term.…”

Section: Introductionmentioning

confidence: 99%

Stability in mean of partial variables for stochastic reaction diffusion systems

Zhang

Luo

Lai

2009

Nonlinear Analysis: Theory, Methods & Applications

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Section: Introductionmentioning

confidence: 99%

Stability in mean of partial variables for stochastic reaction diffusion systems

Zhang

Luo

Lai

2009

Nonlinear Analysis: Theory, Methods & Applications

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“…A different approach to the existence of invariant measures, based on the coupling method, was used by Bogachev and Roechner [2] and C. Mueller [21]. This method can be applied even for space-white noise but only in the case when the space dimension d is one.The existence and uniqueness of the solutions of stochastic reaction-diffusion equations in bounded domains with Dirichlet boundary condition, as well as the existence of an invariant measure was studied by S. Cerrai in [5,4,6] and references therein.The question of the existence of invariant measures in unbounded domains with A = ∆ was studied in [11,13,24,1]. The key condition for the existence of a solution bounded in probability, and hence the existence of an invariant measure in these works is the following dissipation condition for the nonlinearity f : for some k > 0,…”

mentioning

confidence: 99%

“…The question of the existence of invariant measures in unbounded domains with A = ∆ was studied in [11,13,24,1]. The key condition for the existence of a solution bounded in probability, and hence the existence of an invariant measure in these works is the following dissipation condition for the nonlinearity f : for some k > 0,…”

mentioning

confidence: 99%

Existence and Uniqueness of Invariant Measures for Stochastic Reaction–Diffusion Equations in Unbounded Domains

2015

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In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type du = (Au + f (u))dt + σ(u)dW (t), where A is an elliptic operator, f and σ are nonlinear maps and W is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function f possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions f and σ and elliptic operators A for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if A is the Shrödinger-type operator A = 1 ρ (divρ∇u) where ρ = e −|x| 2 is the Gaussian weight.

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“…In ref. [23], the invariant measure of stochastic heat equation was studied by comparison method. Ref.…”

Section: Introductionmentioning

confidence: 99%

Theory and application of stability for stochastic reaction diffusion systems

Luo

Deng

Mao

et al. 2008

Sci. China Ser. F-Inf. Sci.

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So far, the Lyapunov direct method is still the most effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding Itô formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the Itô stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probability, and exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results obtained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper

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Invariant measures for stochastic heat equations with unbounded coefficients

Cited by 21 publications

References 7 publications

Stability in mean of partial variables for stochastic reaction diffusion systems

Stability in mean of partial variables for stochastic reaction diffusion systems

Existence and Uniqueness of Invariant Measures for Stochastic Reaction–Diffusion Equations in Unbounded Domains

Theory and application of stability for stochastic reaction diffusion systems

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