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Recently, a number of authors have investigated the conditions under which a stochastic perturbation acting on an infinite dimensional dynamical system, e.g. a partial differential equation, makes the system ergodic and mixing. In particular, one is interested in finding minimal and physically natural conditions on the nature of the stochastic perturbation. I shall review recent results on this question; in particular, I shall discuss the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity $ u$, and grows like $ u^{-3}$ when $ u$ goes to zero. This Markov process has a unique invariant measure and is exponentially mixing in time.
In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, cor
We study the following equation begin{equation*} frac{partial u(t,,x)}{partial t}= Delta u(t,,x)+b(u(t,,x))+sigma dot{W}(t,,x),quad t>0, end{equation*} where $sigma$ is a positive constant and $dot{W}$ is a space-time white noise. The initial conditi
One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous
A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic part
It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada-Watanabe theorem cite{yamada1,yamada2} and the Feller test for explosions cite{