Gradient bounds for Kolmogorov type diffusions

Baudoin, Fabrice; Gordina, Maria; Mariano, Phanuel

doi:10.1214/19-aihp975

Cited by 8 publications

(11 citation statements)

References 53 publications

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“…where F : R d → R r is a C 1 -function. Similar diffusions were considered in [4]. These processes have infinitesimal generator…”

Section: Wang's Harnack Inequality For Finite-dimensional Diffusionsmentioning

confidence: 96%

“…Before further describing the structure of the paper and the main results, let us mention several papers most relevant to the techniques we are using. In [4] the first two authors and Ph. Mariano study gradient bounds and other related functional inequalities for Kolmogorov-type diffusions in finite dimensions via both the generalized Γ-calculus techniques employed in the present paper as well as through coupling methods.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-invariance for infinite-dimensional Kolmogorov diffusions

Baudoin,

Gordina,

Melcher

2021

Preprint

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We prove Cameron-Martin type quasi-invariance results for the heat kernel measure of infinite-dimensional Kolmogorov and related diffusions. We first study quantitative functional inequalities for appropriate finite-dimensional approximations of these diffusions, and we prove these inequalities hold with dimension-independent coefficients. Applying an approach developed in [5, 10,11], these uniform bounds may then be used to prove that the heat kernel measure for certain of these infinite-dimensional diffusions is quasi-invariant under changes of the initial state. Contents 1. Introduction 1 2. Wang's Harnack inequality for finite-dimensional diffusions 4 2.1. Kolmogorov diffusions 4 2.2. Generalized Kolmogorov diffusions 6 3. Infinite-dimensional results 13 3.1. Infinite-dimensional Kolmogorov diffusion 15 3.2. Generalized Kolmogorov diffusions 18 References 26 Primary 60J60, 28C20; Secondary 35H10

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“…where F : R d → R r is a C 1 -function. Similar diffusions were considered in [4]. These processes have infinitesimal generator…”

Section: Wang's Harnack Inequality For Finite-dimensional Diffusionsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Quasi-invariance for infinite-dimensional Kolmogorov diffusions

Baudoin,

Gordina,

Melcher

2021

Preprint

Self Cite

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“…Then, a strategy for coupling such processes consists in coupling the driving noises and see the effects on the "driven processes". It gives numerous examples for couplings, successful or not [22,10,16,12,28,27,26,3,4,17,3,5]. This strategy can be used to study the subelliptic Brownian motion B t = (X t , z t ) for three model spaces of subRiemannian manifolds:…”

Section: Motivationsmentioning

confidence: 99%

“…In particular, under the above condition x = x ′ , we have the equivalence d cc (g, g ′ ) ∼ |z − z ′ | (see relation (10) in section 2.1). This leads to inequality (4).…”

mentioning

confidence: 96%

Couplings of Brownian Motions on $$SU(2,\mathbb {C})$$

Bénéfice,

Arnaudon,

Bonnefont

2023

Lecture Notes in Computer Science

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In this article we continue the study of couplings of subelliptic Brownian motions on the subRiemannian manifolds SU (2) and SL(2, R). Similar to the case of the Heisenberg group, this subelliptic Brownian motion can be considered as a Brownian motion on the sphere (resp. the hyperbolic plane) together with its swept area modulo 4π. Using this structure, we construct an explicit non co-adapted successful coupling on SU (2) and, under strong conditions on the starting points, on SL(2, R) too. This strategy uses couplings of Brownian bridges, taking inspiration into the work from Banerjee, Gordina and Mariano [3] on the Heisenberg group. We prove that the coupling rate associated to these constructions is exponentially decreasing in time and proportionally to the subRiemannian distance between the starting points. We also give some gradient inequalities that can be deduced from the estimation of this coupling rate.

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“…Similarly, there is a family of diffusions which all have the same small-time fluctuations for the bridge as the iterated Kolmogorov diffusion, that is, a standard Brownian motion together with a finite number of its iterated time integrals. Banerjee and Kendall [2] study maximal and efficient couplings for iterated Kolmogorov diffusions, and Baudoin, Gordina and Mariano [4] obtain gradient bounds for this hypoelliptic diffusion. We close by explicitly determining the small-time fluctuations for the bridge of an iterated Kolmogorov diffusion.…”

Section: 3mentioning

confidence: 99%

Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander type

Habermann¹

2019

Electron. J. Probab.

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We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from 0 to 0 in time 1 of an iterated Kolmogorov diffusion.

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Gradient bounds for Kolmogorov type diffusions

Cited by 8 publications

References 53 publications

Quasi-invariance for infinite-dimensional Kolmogorov diffusions

Quasi-invariance for infinite-dimensional Kolmogorov diffusions

Couplings of Brownian Motions on $$SU(2,\mathbb {C})$$

Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander type

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