The Fokker-Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker-Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time-discrete, iterative variational scheme whose solutions converge to the solution of the Fokker-Planck equation. The major novelty of this iterative scheme is that the time step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker-Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.

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International audienceWe consider a discrete elliptic equation with random coefficients $A$, which (to fix ideas) are identically distributed and independent from grid point to grid point $x\in\mathbb{Z}^d$. On scales large w.\ r.\ t.\ the grid size (i.\ e.\ unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients $A_{hom}$ are characterized by % $$ \xi\cdot A_{hom}\xi\;=\;\langle\left((\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)\right)(0)\rangle, \quad\xi\in\mathbb{R}^d, $$ % where the random field $\phi$ is the unique stationary solution of the ''corrector problem'' % $$ -\nabla\cdot A(\xi+\nabla\phi)\;=\;0 $$ % and $\langle\cdot\rangle$ denotes the ensemble average. \medskip It is known (''by ergodicity'') that the above ensemble average of the energy density $e=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of $e$ on length scales $L$ is estimated as follows: % $$ {\rm var}\left[\sum_{x\in\mathbb{Z}^d}\eta_L(x)\,e(x)\right] \;\lesssim\;L^{-d}, $$ % where the averaging function (i.\ e.\ $\sum_{x\in\mathbb{Z}^d}\eta_L(x)=1$, ${\rm supp}\eta_L\subset[-L,L]^d$) has to be smooth in the sense that $|\nabla\eta_L|\lesssim L^{-1}$. In two space dimensions (i.\ e.\ $d=2$), there is a logarithmic correction. \medskip In other words, smooth averages of the energy density $e$ behave like as if $e$ would be independent from grid point to grid point (which it is not for $d>1$). This result is of practical significance, since it allows to estimate the error when numerically computing $A_{hom}$

We study the effective large-scale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w. r. t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i. e. for the "random environment as seen from a random walker"). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension d > 2) and optimal estimates for regularized versions of the corrector (in dimensions d ≥ 2). We also give a self-contained proof for a new estimate on the gradient of the parabolic, variable-coefficient Green's function, which is a crucial analytic ingredient in our method.As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative methods, we develop an error analysis that gives optimal bounds in terms of scaling in the size of the representative volume element -even for large ellipticity ratios.