Reflection couplings and contraction rates for diffusions

Eberle, Andreas

doi:10.1007/s00440-015-0673-1

Cited by 188 publications

(338 citation statements)

References 38 publications

Supporting

Mentioning

326

Contrasting

Order By: Relevance

“…If the drift vector field b fulfills certain dissipative properties, this latter method provides explicit rate of convergence to equilibrium in a straightforward way, see e.g. , and the preprint . The present work is motivated by , where the author obtained exponential decay of

{(P_{t})}_{t \geq 0}

when the drift b is assumed to be only dissipative at infinity, see the introduction below for more details.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, it follows from [, Theorem 1] or [, Remark 3.6] (also see [, Section 3.1.2, Theorem 1]) that holds for any probability measures μ and ν if and only if holds for all x ,

y \in {double-struckR}^{d}

. The first breakthrough to get rid of this restrictive condition was done recently by Eberle in , at the price of multiplying a constant

C \geq 1

on the right hand side of . To state the main result in , we need the following notation which measures the dissipativity of the drift b :

κ (r) : = sup \{\frac{⟨ σ^{- 1} (x - y), σ^{- 1} (b (x) - b (y)) ⟩}{2 | σ^{- 1} (x - y) |} : x, y \in R^{d} with | σ^{- 1} (x - y) | = r\} .

As in [, (2.3)], we shall assume throughout the paper that

\int_{0}^{s} κ^{+} (r) d r < + \infty for all s > 0 .

This technical condition will be used in Section to construct the auxiliary function.…”

Section: Introductionmentioning

confidence: 99%

“…The first breakthrough to get rid of this restrictive condition was done recently by Eberle in , at the price of multiplying a constant

C \geq 1

on the right hand side of . To state the main result in , we need the following notation which measures the dissipativity of the drift b :

κ (r) : = sup \{\frac{⟨ σ^{- 1} (x - y), σ^{- 1} (b (x) - b (y)) ⟩}{2 | σ^{- 1} (x - y) |} : x, y \in R^{d} with | σ^{- 1} (x - y) | = r\} .

As in [, (2.3)], we shall assume throughout the paper that

\int_{0}^{s} κ^{+} (r) d r < + \infty for all s > 0 .

This technical condition will be used in Section to construct the auxiliary function. Theorem Suppose that the vector field b is locally Lipschitz continuous, and there is a constant

c > 0

such that

κ (r) \leq - c r for all r > 0 large enough .

Then there exist positive constants C ,

λ > 0

such that for any

t > 0

and

μ, ν \in {scriptP}_{1} (R^{d})

, …”

Section: Introductionmentioning

confidence: 99%

“…In particular, when

σ = Id

, the condition holds true if is satisfied only for large

| x - y |

, that is,

⟨ b (x) - b (y), x - y ⟩ \leq {- K | x - y |}^{2}, | x - y | \geq L

for some constant

L > 0

large enough. Therefore, [, Corollary ] implies that the map

μ \mapsto μ P_{t}

converges exponentially with respect to the standard L 1 ‐Wasserstein distance W 1 under locally non‐dissipative drift, see [, Example 1.1] for more details. The proof of [, Corollary ] is based on the coupling by reflection of diffusion processes and a carefully constructed concave function, cf.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, [, Corollary ] implies that the map

μ \mapsto μ P_{t}

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

Luo

Wang

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

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 L>0, then there is a constant λ:=λ(K1,K2,L)>0 such that for all p∈[1,∞), t>0 and x,y∈double-struckRd, Wp(δxPt,δyPt)≤Ce−λt/pright|x−y|1/p,rightif4.pt0.28em|x−y|≤1;right|x−y|,rightif4.pt0.28em|x−y|>1where C:=C(K1,K2,L,p) is a positive constant. This improves the main result in where the exponential convergence is only proved for the L1‐Wasserstein distance.

show abstract

{(P_{t})}_{t \geq 0}

when the drift b is assumed to be only dissipative at infinity, see the introduction below for more details.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, it follows from [, Theorem 1] or [, Remark 3.6] (also see [, Section 3.1.2, Theorem 1]) that holds for any probability measures μ and ν if and only if holds for all x ,

y \in {double-struckR}^{d}

. The first breakthrough to get rid of this restrictive condition was done recently by Eberle in , at the price of multiplying a constant

C \geq 1

on the right hand side of . To state the main result in , we need the following notation which measures the dissipativity of the drift b :

κ (r) : = sup \{\frac{⟨ σ^{- 1} (x - y), σ^{- 1} (b (x) - b (y)) ⟩}{2 | σ^{- 1} (x - y) |} : x, y \in R^{d} with | σ^{- 1} (x - y) | = r\} .

As in [, (2.3)], we shall assume throughout the paper that

\int_{0}^{s} κ^{+} (r) d r < + \infty for all s > 0 .

This technical condition will be used in Section to construct the auxiliary function.…”

Section: Introductionmentioning

confidence: 99%

“…The first breakthrough to get rid of this restrictive condition was done recently by Eberle in , at the price of multiplying a constant

C \geq 1

on the right hand side of . To state the main result in , we need the following notation which measures the dissipativity of the drift b :

κ (r) : = sup \{\frac{⟨ σ^{- 1} (x - y), σ^{- 1} (b (x) - b (y)) ⟩}{2 | σ^{- 1} (x - y) |} : x, y \in R^{d} with | σ^{- 1} (x - y) | = r\} .

As in [, (2.3)], we shall assume throughout the paper that

\int_{0}^{s} κ^{+} (r) d r < + \infty for all s > 0 .

This technical condition will be used in Section to construct the auxiliary function. Theorem Suppose that the vector field b is locally Lipschitz continuous, and there is a constant

c > 0

such that

κ (r) \leq - c r for all r > 0 large enough .

Then there exist positive constants C ,

λ > 0

such that for any

t > 0

and

μ, ν \in {scriptP}_{1} (R^{d})

, …”

Section: Introductionmentioning

confidence: 99%

“…In particular, when

σ = Id

, the condition holds true if is satisfied only for large

| x - y |

, that is,

⟨ b (x) - b (y), x - y ⟩ \leq {- K | x - y |}^{2}, | x - y | \geq L

for some constant

L > 0

large enough. Therefore, [, Corollary ] implies that the map

μ \mapsto μ P_{t}

Section: Introductionmentioning

confidence: 99%

“…Therefore, [, Corollary ] implies that the map

μ \mapsto μ P_{t}

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

Luo

Wang

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

On the Poincaré Constant of Log-Concave Measures

Cattiaux

Guillin

2020

Lecture Notes in Mathematics

View full text Add to dashboard Cite

On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

Delgadino

Gvalani

Pavliotis

2021

Arch Rational Mech Anal

View full text Add to dashboard Cite

The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

show abstract

Reflection couplings and contraction rates for diffusions

Cited by 188 publications

References 38 publications

Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

On the Poincaré Constant of Log-Concave Measures

On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

Contact Info

Product

Resources

About