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THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION
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 ...
THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show
Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Abstract We show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6 , 587–600) for the Gaussian measure, are implied by logarithmic Sobolev
An optimal variance estimate in stochastic homogenization of discrete elliptic equations
We 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
L1-Contraction and Uniqueness for Quasilinear Elliptic–Parabolic Equations
Abstract We prove the L 1 -contraction principle and uniqueness of solutions for quasilinear elliptic–parabolic equations of the form[formula]where b is monotone nondecreasing and continuous. We
Threshold dynamics for networks with arbitrary surface tensions
We present and study a new algorithm for simulating the N-phase mean curvature motion for an arbitrary set of (isotropic) surface tensions. The departure point is the threshold dynamics algorithm of
A Regularity Theory for Random Elliptic Operators
Since the seminal results by Avellaneda & Lin it is known that elliptic operators with periodic coefficients enjoy the same regularity theory as the Laplacian on large scales. In a recent inspiring
An optimal error estimate in stochastic homogenization of discrete elliptic equations
We 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
Upper Bounds on Coarsening Rates
Abstract: We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a
Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics
We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice $$\mathbb Z^d$$Zd with random coefficients. The theory of stochastic homogenization relates
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