The logarithmic Sobolev inequality for the Wasserstein diffusion

Abstract

We prove that the Dirichlet form associated with the Wasserstein diffusion on the set of all probability measures on the unit interval, introduced in von Renesse and Sturm (Entropic measure and Wasserstein diffusion. Ann Probab, 2008) satisfies a logarithmic Sobolev inequality. This implies hypercontractivity of the associated transition semigroup. We also study functional inequalities for related diffusion processes. Mathematics Subject Classification (2000)Primary: 58J65 · 47D07; Secondary: 28A33 · 35P15 · 60J35

Entropic Measure on Multidimensional Spaces

“…The Wasserstein diffusion on 1-dimensional spaces satisfies a logarithmic Sobolev inequality [DS07]; it can be obtained as scaling limit of empirical distributions of interacting particle systems [AR07].…”

Particle approximation of the Wasserstein diffusion

“…The lower semicontinuity ofẼ in L 2 (G, Q β ) is trivial as well as the fact thatẼ is an extension E, due to (8). To prove thatẼ is Markovian we use the stronger quasi-invariance of Q β under certain transformations of G, cf.…”

“…(The pointwise convergence of the same sequence E N to E has been used in a recent work by Döring and Stannat to establish the logarithmic Sobolev inequality for E, cf. [8].) Since the approximating state spaces are finite dimensional we employ [19] for a generalized framework of Mosco-convergence of forms defined on a scale of Hilbert spaces.…”

A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval

“…Of course, none of this precludes an arbitrary reference measures in P(P(M )) giving rise to lower Ricci bounds on P(M ) (in particular, a point mass, say δ Leb ∈ P(P([0, 1])) certainly has such bounds, but is not a particularly interesting example). 3 We remark that the log-Sobolev and Poincaré inequalities are certainly weaker conditions than generalized lower Ricci bounds. To see this, consider the possibility of such inequalities on a metric measure space (X, d, m).…”

“…[8, Corollary 6.12, Theorem 6.18]), (and in particular, Theorem 1.8 would be the consequence of the space having generalized Ricci bounded below by β). 3 However, in spite of these heuristics, there are no such Ricci lower bounds, as we see in Theorem 1.1. We do remark that in [5], Gigli has argued that there is no natural choice of volume form on P(M ), because this would be equivalent to there existing a Laplacian (by an integration by parts formula), which seems not to exist, because of the issues related to tracing a Hessian type object over an infinite dimensional space.…”