Coupling of Multidimensional Diffusions by Reflection

Lindvall, Torgny; Rogers, L. C. G.

doi:10.1214/aop/1176992442

Cited by 151 publications

(175 citation statements)

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“…This is why we regularize also the coefficients of the coupling operator, introducing an additional term, depending on ε > 0; in this way the coefficients of the coupling operators are continuous on the whole space, rather than merely well defined outside the diagonal as it is usual (cf. [5,15]). We only prove the existence of such a family of coupling processes, depending on ε, but not their uniqueness as it happens in other cases of coupling available in the literature (see, e.g., [15]).…”

Section: Introductionmentioning

confidence: 99%

Gradient estimates for diffusion semigroups with singular coefficients

Priola

Wang

2006

Journal of Functional Analysis

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Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the R d -case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of R d with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the R d -case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying diffusion semigroup.

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Section: Introductionmentioning

confidence: 99%

Gradient estimates for diffusion semigroups with singular coefficients

Priola

Wang

2006

Journal of Functional Analysis

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“…From (2.3) the mirror coupling is indeed a maximal coupling, a well-known fact (see Lindvall [10] and Lindvall and Rogers [11]). It was believed that the mirror coupling is the unique maximal coupling of Euclidean Brownian motion.…”

Section: Maximal Couplingmentioning

confidence: 87%

“…For Euclidean Brownian motion, it is well known (Lindvall and Rogers [11]) that the mirror coupling, which we will define shortly, is a maximal coupling.…”

Section: Maximal Couplingmentioning

confidence: 99%

Maximal Coupling of Euclidean Brownian Motions

Hsu

Sturm

2013

Commun. Math. Stat.

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“…The coupling ideas used here originated in Lindvall and Rogers (1986) for the Euclidean case and in Kendall (1986a) in the geometric setting. The development of the latter reference was extended in Cranston (1991) and we now recall these results.…”

Section: Bm(m) and Couplingsmentioning

confidence: 99%