On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Albeverio, Sergio; Богачев, В. И.; Röckner, Michael

doi:10.1002/(sici)1097-0312(199903)52:3<325::aid-cpa2>3.0.co;2-v

Cited by 66 publications

(61 citation statements)

References 24 publications

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“…The above result actually holds in the case of an arbitrary piecewise constant function . In the next section we will see how it can be extended to the general nonlinear feedback system (1). Now suppose that we are given quantizers and linearly independent vectors .…”

Section: Quantized Feedback Systemsmentioning

confidence: 99%

“…Namely, systems that are monotemperaturic in the sense of [11] turn out to be compatible. This shows more clearly the place that compatible systems occupy among all systems of the form (1). In Section VI we single out a class of systems for which the compatibility condition takes a particularly transparent form and show how the steady-state probability densities are related to Lyapunov functions for deterministic nonlinear feedback systems.…”

Section: Introductionmentioning

confidence: 98%

“…In this paper we study Itô stochastic systems of the form (1) Manuscript received March 20, 1998; revised April 30, 1999. Recommended by Associate Editor, Q. Zhang.…”

Section: Introductionmentioning

confidence: 99%

“…The Fokker-Planck operator associated with (1) is given by the formula tr (2) The problem under consideration is that of finding an explicit formula for a steady-state probability density associated with (1). This amounts to solving the steady-state version of the Fokker-Planck equation (3) subject to the constraints and .…”

Section: Introductionmentioning

confidence: 99%

“…As is well known, under the above assumptions the (unique) steady-state probability density is a Gaussian det where is the positive definite symmetric steady-state variance matrix satisfying the equation (4) In this paper we will be concerned with extending this result to nonlinear feedback systems of the form (1). In Section II we study the case when the nonlinearity is given by a piecewise constant function of a certain type, called a quantizer.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

Liberzon

Brockett

2000

IEEE Trans. Automat. Contr.

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Abstract-In this paper we describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steady-state probability densities can be found. We show that this class includes what has been called monotemperaturic systems in earlier work and establish relationships with Lyapunov functions for the corresponding deterministic systems. We also treat a number of stochastic optimal control problems in the case of quantized feedback, with performance criteria formulated in terms of the steady-state probability density.Index Terms-Nonlinear feedback system, quantizer, steady-state probability density, stochastic optimal control, white noise.

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Section: Quantized Feedback Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

“…In this paper we study Itô stochastic systems of the form (1) Manuscript received March 20, 1998; revised April 30, 1999. Recommended by Associate Editor, Q. Zhang.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

Liberzon

Brockett

2000

IEEE Trans. Automat. Contr.

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A Priori Estimates for Symmetrizing Measures and Their Applications to Gibbs States

Albeverio

Kondratiev

Röckner

et al. 2000

Journal of Functional Analysis

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Uniqueness Results for the Generators of the Two-Dimensional Euler and Navier–Stokes Flows

Albeverio¹,

Ferrario

2002

Journal of Functional Analysis

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The Euler and Navier-Stokes equations for an incompressible fluid in two dimensions with periodic boundary conditions are considered. Concerning the Euler equation, previous works analyzed the associated (first order) Liouville operator L as a symmetric linear operator in a Hilbert space L 2 ðm g Þ with respect to a natural invariant Gaussian measure m g (given by the enstrophy), with the domain subspace of cylinder smooth bounded functions and have shown that there exist self-adjoint extensions of L. For the Navier-Stokes equation with a suitable white noise forcing term, the associated (second order) Kolmogorov operator K has been considered on the same domain as the sum of the Liouville operator L with the Ornstein-Uhlenbeck operator Q corresponding to the Stokes operator and the forcing term; existence of a C 0 -semigroup of contraction in L 2 ðm g Þ with generator extending the operator K has been proven. In this paper it is proven that both L and K are bounded by naturally associated positive Schro¨dinger-like operators, which are essentially self-adjoint on a dense subspace of cylinder functions. Other uniqueness results concerning L, respectively, K are also given. # 2002 Elsevier Science (USA)

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On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Cited by 66 publications

References 24 publications

Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

A Priori Estimates for Symmetrizing Measures and Their Applications to Gibbs States

Uniqueness Results for the Generators of the Two-Dimensional Euler and Navier–Stokes Flows

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