Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Wang, Feng‐Yu

doi:10.1007/s11118-019-09800-z

Cited by 21 publications

(23 citation statements)

References 34 publications

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“…In the general settings of Riemannian manifold and of SDEs with multiplicative noises, F.-Y. Wang [27] obtained the exponential decay in the L 2 -Wasserstein distance under B(K 1 r, K 2 r, l 0 ), i.e., (1.3) holds with Φ 1 (r) = K 1 r for some K 1 > 0; moreover, he establishes similar results for the L p -Wasserstein distance for all p ≥ 1 provided that the diffusion semigroup is ultracontractive. Some developments in the jump case can be found in [31,17] under B(K 1 r, K 2 r, l 0 ).…”

Section: )mentioning

confidence: 99%

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Luo

Wang

2019

Stochastic Processes and their Applications

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We establish the exponential convergence with respect to the L 1 -Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equationwhere (Z t ) t≥0 is a pure jump Lévy process whose Lévy measure ν fulfillsfor some constant κ 0 > 0, and the drift term b satisfies that for anywith some positive constants K 2 , l 0 and positive measurable function Φ 1 . The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process (X t ) t≥0 .where Φ 1 and Φ 2 are two nonnegative measurable functions, and l 0 ≥ 0 is a constant. For example, when Φ 2 (r) = K 2 r for some positive constantThis holds if the drift term b is dissipative outside some compact set. In particular, when Φ 1 (r) = K 1 r for some constant K 1 ≥ 0, it follows from (1.3) that for anywhich, along with (1.3), yields that the SDE (1.1) has a non-explosive and pathwise unique strong solution, see [1, Chapter 6, Theorem 6.2.3] (in the standard Lipschitz case) or [11, Theorem 2] and [25, Chapter 3, Theorem 115] (in the one-sided Lipschitz case). Note that, sincewe are sometimes concerned with only measurable drift term b, non-Lipschitz condition like B(K 1 , K 2 r, l 0 ) will also be adopted in our results below. The reader can refer to [8,20,22,26,32] and references therein for recent studies on the existence and uniqueness of strong solution to (1.1) with non-regular drift term. In particular, assuming that Z is the truncated symmetric α-stable process on R d with α ∈ (0, 2), and b is bounded and β-Hölder continuous with β > 1 − α/2, it was proved in [8, Corollary 1.4(i)] that the SDE (1.1) has a unique strong solution for each x ∈ R d . Furthermore, in the one-dimensional case, if α > 1, then the SDE (1.1) also enjoys a unique strong solution for each x ∈ R, even if the drift b is only bounded and measurable (see [26, Remark 1, p. 82]). Denote by ν the Lévy measure of the pure jump Lévy process Z. We assume that there is a constant κ 0 > 0 such that(1.4)Condition (1.4) was first used in [23] to study the coupling property of Lévy processes. It is satisfied by a large class of Lévy measures. For instance, iffor some z 0 ∈ R d and some ε > 0 such that ρ 0 (z) is positive and continuous on B(z 0 , ε), then such Lévy measure ν fulfills (1.4), see [24, Proposition 1.5] for details. Actually, as shown in Proposition 6.5, the condition (1.4) implies that there is a nonnegative measurable function ρ on R d such that ν(dz) ≥ ρ(z) dz andLet (P t ) t≥0 be the transition semigroup associated with the process (X t ) t≥0 . In this paper we are interested in the asymptotics of the Wasserstein-type distances (including the L 1 -Wasserstein distance and the total variation) between probability distributions δ x P t = P t (x, ·) and δ y P t = P t (y, ·) for any x, y ∈ R d , when the drift term b is dissipative outside some compact set, i.e. b satisfies B(Φ 1 (r), K 2 r, l 0 ) for some positive measurable function Φ 1 , and some const...

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Section: )mentioning

confidence: 99%

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Luo

Wang

2019

Stochastic Processes and their Applications

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“…Coupling for SDEs driven by multiplicative Brownian motions is a well developed field, and there is a vast literature on this topic; we mention here the papers [6,11,20,25] and the monographs [5,10,23,24]. Notice that, in contrast with the case of coupling for SDEs driven by multiplicative Brownian motions, our case for multiplicative Lévy noises is quite different.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For example, we use coupling by reflection for (B ′ t ) t≥0 and coupling by parallel displacement for (B ′′ t ) t≥0 . Usually the term of coupling by reflection for (B ′ t ) t≥0 plays a leading role in applications, see [20,25]. However, such nice additive property fails if we apply to the SDE (1.1), i.e., replace (B t ) t≥0 by the Lévy process (Z t ) t≥0 in the argument above.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

Liang

Wang

2020

Stochastic Processes and their Applications

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We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to α-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the L 1 -Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which is derived for the first time in this paper.

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“…If the LSI holds, we have κ = 1/C LS . Very recently, Wang [19] discussed exponential contraction in any W p (p 1) for a class of diffusion semigroups and gave the implication from (2) to (1) as well.…”

Section: Introductionmentioning

confidence: 99%

The Poincaré inequality and quadratic transportation-variance inequalities

Liu¹

2020

Electron. J. Probab.

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It is known that the Poincaré inequality is equivalent to the quadratic transportation-variance inequality (namely Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller C V than before, which equals the double of Poincaré constant. Applying the same argument leads to more characterizations of the Poincaré inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Émery curvature has a lower bound (here the control constants may depend on the curvature bound).Next, we present a comparison inequality between W 2 2 (f µ, µ) and its centralization W 2 2 (fcµ, µ), which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of W 2 2 (f µ, µ) associated to the variance of √ f rather than f . As a by-product, we have another proof to derive the quadratic transportationinformation inequality from Lyapunov condition, avoiding the Bobkov-Götze's characterization of the Talagrand's inequality. π∈C(ν,µ) E×E d p (x, y)π(dx, dy) 1/p ,

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Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Cited by 21 publications

References 34 publications

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

The Poincaré inequality and quadratic transportation-variance inequalities

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