8,172 research outputs found
Rigorous results in space-periodic two-dimensional turbulence
We survey the recent advance in the rigorous qualitative theory of the 2d
stochastic Navier-Stokes system that are relevant to the description of
turbulence in two-dimensional fluids. After discussing briefly the
initial-boundary value problem and the associated Markov process, we formulate
results on the existence, uniqueness and mixing of a stationary measure. We
next turn to various consequences of these properties: strong law of large
numbers, central limit theorem, and random attractors related to a unique
stationary measure. We also discuss the Donsker-Varadhan and Freidlin-Wentzell
type large deviations, as well as the inviscid limit and asymptotic results in
3d thin domains. We conclude with some open problems
Kac's chaos and Kac's program
In this note I present the main results about the quantitative and
qualitative propagation of chaos for the Boltzmann-Kac system obtained in
collaboration with C. Mouhot in \cite{MMinvent} which gives a possible answer
to some questions formulated by Kac in \cite{Kac1956}. We also present some
related recent results about Kac's chaos and Kac's program obtained in
\cite{MMWchaos,HaurayMischler,KleberSphere} by K. Carrapatoso, M. Hauray, C.
Mouhot, B. Wennberg and myself
Stochastic stability and the design of feedback controls
Stochastic stability and design of feedback controls - application of Liapunov method to optimal control problem
Limits of relative entropies associated with weakly interacting particle systems
The limits of scaled relative entropies between probability distributions
associated with N-particle weakly interacting Markov processes are considered.
The convergence of such scaled relative entropies is established in various
settings. The analysis is motivated by the role relative entropy plays as a
Lyapunov function for the (linear) Kolmogorov forward equation associated with
an ergodic Markov process, and Lyapunov function properties of these scaling
limits with respect to nonlinear finite-state Markov processes are studied in
the companion paper [6]
Stochastic stability
The field of stochastic stability is surveyed, with emphasis on the invariance theorems and their potential application to systems with randomly varying coefficients. Some of the basic ideas are reviewed, which underlie the stochastic Liapunov function approach to stochastic stability. The invariance theorems are discussed in detail
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