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Register a new userErgodicity and mixing for stochastic partial differential equations
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Jean Bricmont
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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.
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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, corresponding to a large number of ``particles (or ``agents). The objective of the present paper is to deepen the investigation of such Mean-Field BSDEs by studying them in a more general framework, with general driver, and to discuss comparison results for them. In a second step we are interested in partial differential equations (PDE) whose solutions can be stochastically interpreted in terms of Mean-Field BSDEs. For this we study a Mean-Field BSDE in a Markovian framework, associated with a Mean-Field forward equation. By combining classical BSDE methods, in particular that of ``backward semigroups introduced by Peng [14], with specific arguments for Mean-Field BSDEs we prove that this Mean-Field BSDE describes the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated to Mean-Field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions.
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 condition $u(0,x)=u_0(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman, our first result shows that the solution blows up in finite time if and only if begin{equation*} int_{cdot}^inftyfrac{1}{b(s)},d s<infty, end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $mathbb{R}^d$.
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 measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations. Here, we develop a similar theory for some stochastic partial differential equations.
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 partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.
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{feller51,feller54}, there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called {it free energy function} on the density function space, we prove that the density functions, which are solutions of the Fokker-Planck equation, converge to the stationary density function exponentially under the Kullback-Leibler {divergence}, thus also in the total variation norm. The results show that there is a relation between the Bakry-Emery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher-Lamperti transformation. Several applications are discussed, including the Cox-Ingersoll-Ross model and the Ait-Sahalia model in finance and the Wright-Fisher model in population genetics.
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