Maximal Coupling of Euclidean Brownian Motions

Hsu, Erin L.; Sturm, Karl Theodor

doi:10.1007/s40304-013-0007-5

Cited by 32 publications

(53 citation statements)

References 11 publications

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“…Straightforward calculations, based on the reflection principle, show that this construction is in fact a Markovian maximal coupling (MMC). Furthermore, [18] proved that this is the unique such coupling for Euclidean Brownian motion. A few other examples are discussed in the literature:…”

Section: Introductionmentioning

confidence: 90%

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Rigidity for Markovian maximal couplings of elliptic diffusions

Banerjee

Kendall

2016

Probab. Theory Relat. Fields

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Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm, in a preprint circulating in 2007, but later published in 2013, proved that the reflection-coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada (Electron J Probab 14(25), 2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups). This permits a complete characterization of those possible manifolds and their diffusions for which there exists

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Section: Introductionmentioning

confidence: 90%

“…Since X τ = Y τ , we can extend X and Y synchronously beyond time τ . Combined with (18), this implies τ = τ almost surely, since the maximal coupling time τ must be stochastically smaller than all other coupling times. Consequently…”

Section: Corollary 12 Consider a Markovian Maximal Coupling With Coumentioning

confidence: 99%

Rigidity for Markovian maximal couplings of elliptic diffusions

Banerjee

Kendall

2016

Probab. Theory Relat. Fields

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“…Thus, for any fixed z ∈ R d , (x, y) → R x,y (z) is locally Lipschitz continuous on {(x, y) ∈ R 2d : x = y}. If we assume, in addition, that the drift term b is Lipschitz continuous, then the SDE (11) has a unique strong solution (X t , Y t ) up to t < τ , where τ is the coupling time defined by (12) τ := inf{t > 0 :…”

Section: 1mentioning

confidence: 99%

“…It has been efficiently used to show regularity properties of Markov semigroups and ergodicity of Markov processes. There are many publications on coupling of diffusion processes, see for instance [18,9,24,12,3] and the references therein, but only few papers consider the coupling of jump processes. The first systematic investigations on coupling of Lévy processes are [27,5,26] and [6, Chapter 6.2], but -compared to the diffusion case -the theory is still in its infancy.…”

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confidence: 99%

A unified approach to coupling SDEs driven by Lévy noise and some applications

Liang¹,

Schilling²,

Wang³

2020

Bernoulli

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We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive Lévy noise.2010 Mathematics Subject Classification. 60J35; 60G51; 60H10; 60J25; 60J75.

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“…To construct an optimal Markov coupling process of subordinate Brownian motion (X t ) t≥0 , we begin with reviewing known facts about the coupling of Brownian motions by reflection, see [12,3,10]. Fix x, y ∈ R d with x ̸ = y.…”

Section: Couplings Of Subordinate Brownian Motionsmentioning

confidence: 99%