Stochastic Dissipative PDE's and Gibbs Measures

By Sergei B. Kuksin, Armen Shirikyan. 2000

162 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8d1556a52f3a8257d27faee12fbe21a4a80c12f8

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Citations in the corpus

Statistical mechanics of two-dimensional and geophysical flows (2011, background): “In the case α = 0, many interesting mathematical results have been recently obtained for the stochastic Navier–Stokes equations (91): the existence of an invariant measure, its properties in the Euler limit ν → 0, the validity of the law of large numbers, central limit theorems, ergodicity (see [115”
Ergodicity for the Randomly Forced 2D Navier–Stokes Equations (2001, methodology): “See the introduction to [KS1] for discussion of this result and see [G, KS1 ] for its relations with statistical hydrodynamics.”
Exponential mixing for stochastic PDEs: the non-additive case (2005, background): “Hence H1 can be seen as a non degeneracy condition on the low modes in the spirit of [3], [8], [9], [14], [19], [21], [22], [23], [26], [30] and [31].”
A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs (2008, methodology): “The technique of proof can be interpreted as an “infinitesimal” version of techniques developed in [EMS0 1, KS00 ] and extended in [BKL01, MY02, Mat02, Hai02, BM05] combined with detailed lower bounds on the Malliavin covariance matrix of the solution.”
Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation (2001, background): “Recently Kuksin and Shirikyan [KS] proved uniqueness of stationary measure when the Navier-Stokes equation is perturbed by a bounded degenerate kicked noise.”
Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics (2002, methodology): “In this paper, we take a more probabilistic approach than [ KS00 , KS, BKL01].”
Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation (2001, unknown): “In fact, for the full Navier-Stokes equation, the only existing result so far for the uniqueness of an invariant measure under degenerate noise is for the case when all determining modes are forced [2, 4, 14].”
Ergodic Theory of Infinite Dimensional Systems¶with Applications to Dissipative Parabolic PDEs (2002, background): “(In this regard, we are influenced by [7], which takes a similar approach.”