Duality and hypoellipticity: one-dimensional case studies

Miclo, Laurent

doi:10.1214/17-ejp114

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…Of course Corollary 7 and Proposition 65 are well-known in the present elliptic diffusion framework. Nevertheless, we think this new approach can be adapted to more complicated context, as Theorem 5 is quite universal (it was shown to hold also for hypoelliptic diffusions, for the moment in dimension 1, in [18]). We believe it should always be possible to associate to a diffusion some evolving sets (as mentioned in the introduction) whose weights for an invariant measure behave like a continuous martingale.…”

Section: Proofmentioning

confidence: 99%

“…Here we make an important step further in this program, by showing that elliptic diffusions on differential manifolds admitting an invariant measure can indeed be intertwined with domain-valued Markov processes. Although the hypoellipticity is not yet entering in the game (but see [18] for a first illustration in dimension 1), the introduced domain-valued processes are already very intriguing and promising for themselves. When dealing with the Brownian motion on a Riemannian manifold, they are natural stochastic modifications of the mean curvature flow.…”

Section: Introductionmentioning

confidence: 99%

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On the evolution by duality of domains on manifolds

Coulibaly-Pasquier

Miclo

2021

msmf

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On a manifold, consider an elliptic diffusion X admitting an invariant measure µ. The goal of this paper is to introduce and investigate the first properties of stochastic domain evolutions pD t q tPr0,τs which are intertwining dual processes for X (where τ is an appropriate positive stopping time before the potential emergence of singularities). They provide an extension of Pitman's theorem, as it turns out that pµpD t qq tPr0,τs is a Bessel-3 process, up to a natural time-change. When X is a Brownian motion on a Riemannian manifold, the dual domain-valued process is a stochastic modification of the mean curvature flow to which is added an isoperimetric ratio drift to prevent it from collapsing into singletons. RésuméSur une variété, considérons une diffusion elliptique X de mesure invariante µ. Le but de ce papier est d'introduire et d'étudier les premières propriétés d'évolutions stochastiques de domaines pD t q tPr0,τs qui sont des processus duaux par entrelacement de X (où τ est un temps d'arrêt strictement positif précédant l'apparition éventuelle de singularités). Il s'agit d'une extension du théorème de Pitman, puisqu'il ressort que pµpD t qq tPr0,τs est un processus de Bessel-3, à un changement naturel de temps près. Quand X est un mouvement brownien sur une variété compacte, ce processus dual à valeurs domaines est une modification stochastique du flot par courbure moyenne auquel est ajouté une dérive fournie par un quotient isopérimétrique qui l'empêche de s'effondrer en des singletons.

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Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the evolution by duality of domains on manifolds

Coulibaly-Pasquier

Miclo

2021

msmf

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Strong stationary times for finite Heisenberg walks

Miclo¹

2023

ESAIM: PS

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The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over $\ZZ_M$, for odd $M\geq 3$. When they correspond to $3\times 3$ matrices, the strong stationary times are of order $M^6$, estimate which can be improved to $M^4$ if we are only interested in the convergence to equilibrium of the last column. Simulations by Chhaïbi suggest that the proposed strong stationary time is of the right $M^2$ order. These results are extended to $N\times N$ matrices, with $N\geq 3$. All the obtained bounds are thought to be non-optimal, nevertheless this original approach is promising, as it relates the investigation of the previously elusive strong stationary times of such random walks to new absorbing Markov chains with a statistical physics flavor and whose quantitative study is to be pushed further. In addition, for $N=3$, a strong equilibrium time is proposed in the same spirit for the non-Markovian coordinate in the upper right corner. This result would extend to separation discrepancy the corresponding fast convergence for this coordinate in total variation and open a new method for the investigation of this phenomenon in higher dimension.

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On the construction of measure-valued dual processes

Miclo¹

2020

Electron. J. Probab.

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Duality and hypoellipticity: one-dimensional case studies

Cited by 3 publications

References 6 publications

On the evolution by duality of domains on manifolds

On the evolution by duality of domains on manifolds

Strong stationary times for finite Heisenberg walks

On the construction of measure-valued dual processes

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