Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
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Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature Feng-Yu Wang1,2 Received: 28 May 2017 / Accepted: 9 September 2019 / © Springer Nature B.V. 2020
Abstract Let Pt be the (Neumann) diffusion semigroup Pt generated by a weighted Laplacian on a complete connected Riemannian manifold M without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant λ > 0 if and only if Wp (μ1 Pt , μ2 Pt ) ≤ e−λt Wp (μ1 , μ2 ), t ≥ 0, p ≥ 1 holds for all probability measures μ1 and μ2 on M, where Wp is the Lp Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction Wp (μ1 Pt , μ2 Pt ) ≤ ce−λt Wp (μ1 , μ2 ), p ≥ 1, t ≥ 0 for some constants c, λ > 0 for a class of diffusion semigroups with negative curvature where the constant c is essentially larger than 1. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise. Keywords Wasserstein distance · Diffusion semigroup · Riemannian manifold · Curvature condition · SDEs with multiplicative noise Mathematics Subject Classification (2010) 60J75 · 47G20 · 60G52
1 Introduction Let M be a d-dimensional connected complete Riemannian manifold possibly with a convex boundary ∂M. Let ρ be the Riemannian distance. Consider L = + Z for the LaplaceBeltrami operator and some C 1 -vector field Z such that the (reflecting) diffusion process Supported in part by NNSFC (11771326, 11831014) Feng-Yu Wang
[email protected]; [email protected] 1
Laboratory of Mathematical and Complex Systems, Beijing Normal University, Beijing, 100875, China
2
Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK
F.-Y. Wang
generated by L is non-explosive. Then the associated Markov semigroup Pt is the (Neumann if ∂M = ∅) semigroup generated by L on M. In particular, it is the case when the curvature of L is bounded below; that is, RicZ := Ric − ∇Z ≥ K
(1.1)
holds for some constant K ∈ R. Here and throughout the paper, we write T ≥ h for a (not necessarily symmetric) 2-tensor T and a function h provided
T (X, X) ≥ h(x)|X|2 , X ∈ Tx M, x ∈ M. There exist many inequalities on Pt which are equivalent to the curvature condition (1.1), see [5, 18, 21, 35] for details. In particular, for any constant K ∈ R, the Wasserstein distance inequality Wp (μ1 Pt , μ2 Pt ) ≤ e−Kt Wp (μ1 , μ2 ), t ≥ 0, p ≥ 1, μ1 , μ2 ∈ P (M)
(1.2)
is equivalent to the curvature condition (1.1). Here, P (M) is the class of all probability measures on M; Wp is the Lp -Warsserstein distance induced by ρ, i.e., Wp (μ1 , μ2 ) :=
inf
π∈C (μ1 ,μ2 )
ρLp (π) , μ1 , μ2 ∈ P (M),
where C (μ1 , μ2 ) is the class of all couplings of μ1 and μ2 ; and for a Markov operator P on Bb (M) (i.e. P is a positivity-preserving linear operator with P 1 = 1),
(νP )(A) := ν(P 1A ), A ∈ B (M), ν ∈ P (M),
where ν(f ) := M f dν for f ∈ L1 (ν). In some references, νP is als
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