2016
DOI: 10.1002/mana.201500351
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Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

Abstract: By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups (Pt)t≥0 with respect to the standard Lp‐Wasserstein distance for all p∈[1,∞). In particular, we show that for the Itô stochastic differential equation dXt=dBt+b(Xt)dt,if the drift term b is such that for any x,y∈double-struckRd, ⟨b(x)−b(y),x−y⟩≤rightK1|x−y|2,right|x−y|≤L;right−K2|x−y|2,right|x−y|>Lholds with some positive constants K1, K2 and … Show more

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Cited by 39 publications
(52 citation statements)
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References 31 publications
(61 reference statements)
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“…where C > 0 is a positive constant depending on θ, K 1 , K 2 , L 0 and p. Theorem 1.2 above does provide new conditions on the drift term b such that the associated semigroup (P t ) t≥0 is exponentially contractive with respect to the L p -Wasserstein distance W p for all p ≥ 1. In particular, when α ∈ (1, 2), the conclusion of Theorem 1.2 is the same as that of [15], Theorem 1.3, for diffusion processes; while for α ∈ (0, 1] we need the restrictive condition (1.5); see Remark 3.3 for a further comment. Indeed, (1.5) is natural in the sense that, when α ∈ (0, 1] the drift term plays the dominant role or the same role (just in case that α = 1) for the behavior of SDEs driven by symmetric α-stable processes, see, for example, [2,9] for (Dirichlet) heat kernel estimates and [24] for dimensional free Harnack inequalities on this topic.…”
Section: )mentioning
confidence: 82%
See 1 more Smart Citation
“…where C > 0 is a positive constant depending on θ, K 1 , K 2 , L 0 and p. Theorem 1.2 above does provide new conditions on the drift term b such that the associated semigroup (P t ) t≥0 is exponentially contractive with respect to the L p -Wasserstein distance W p for all p ≥ 1. In particular, when α ∈ (1, 2), the conclusion of Theorem 1.2 is the same as that of [15], Theorem 1.3, for diffusion processes; while for α ∈ (0, 1] we need the restrictive condition (1.5); see Remark 3.3 for a further comment. Indeed, (1.5) is natural in the sense that, when α ∈ (0, 1] the drift term plays the dominant role or the same role (just in case that α = 1) for the behavior of SDEs driven by symmetric α-stable processes, see, for example, [2,9] for (Dirichlet) heat kernel estimates and [24] for dimensional free Harnack inequalities on this topic.…”
Section: )mentioning
confidence: 82%
“…The first breakthrough to get rid of such restrictive condition in this direction for L 1 -Wasserstein distance W 1 was done recently by Eberle in [10,11], at the price of multiplying a constant C ≥ 1 on the right-hand side of (1.3). See [10], Corollary 2.3, for more details, and [15], Theorem 1.3, for related developments on L p -Wasserstein distance W p with all p ∈ [1, ∞) on this topic. However, the corresponding result for SDEs driven by Lévy noises is not available yet now.…”
mentioning
confidence: 99%
“…Step 2. Based on (3.4), the proof of the desired assertion (3.3) is similar to that of [16,Theorem 1.3] or [31, Theorem 1.2] by some slight modifications. For the sake of completeness, we present the details here.…”
Section: )mentioning
confidence: 97%
“…We intend to investigate the W p -exponential contraction for p ∈ [1, ∞). As mentioned in Introduction that existing results only apply to p = 1 and σ = I, and as mentioned in [11,17] that there is essential difficulty to prove (1.3) for p > 1 even for σ = I. So, the present study is non-trivial.…”
Section: Sdes With Multiplicative Noisementioning
confidence: 81%