Convergence of approximations of monotone gradient systems

Abstract

We consider stochastic differential equations in a Hilbert space, perturbed by the gradient of a convex potential. We investigate the problem of convergence of a sequence of such processes. We propose applications of this method to reflecting O.U. processes in infinite dimension, to stochastic partial differential equations with reflection of Cahn-Hilliard type and to interface models.

Existence and stability for Fokker–Planck equations with log-concave reference measure

“…In the next two propositions, borrowed essentially from [36], we show that, for convex functions U : R k → R, the growth at infinity of ∇U is always balanced by the factor e −U ; this leads to uniform bounds and tightness estimates for the measures |∇U |e −U L k , under uniform lower bounds on U .…”

“…In the next two propositions, borrowed essentially from [36], we show that, for convex functions U : R k → R, the growth at infinity of ∇U is always balanced by the factor e −U ; this leads to uniform bounds and tightness estimates for the measures |∇U |e −U L k , under uniform lower bounds on U . Proposition A.2 Let U : R k → R ∪ {+∞} be convex and lower semicontinuous, with U (x) → +∞ as x → +∞, {U < +∞} having a nonempty interior, and set γ = exp(−U )L k .…”

“…Our Theorem 1.5 extends the convergence result to more general initial conditions and, being based only on the log-concavity of the invariant measures, can be applied to a large class of models. For a different (and weaker) approach based on infinite dimensional integration by parts, see [35] and [36].…”