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On the evolution by duality of domains on manifolds

Author

Listed:
  • Abdoulaye Koléhè Coulibaly-Pasquier

    (IECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche Scientifique)

  • Laurent Miclo

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

Abstract

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 (Dt)t∈[0,τ] 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 (μ(Dt))t∈[0,τ] 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.

Suggested Citation

  • Abdoulaye Koléhè Coulibaly-Pasquier & Laurent Miclo, 2022. "On the evolution by duality of domains on manifolds," Post-Print hal-03671574, HAL.
    • Handle: RePEc:hal:journl:hal-03671574
    DOI: 10.24033/msmf.479
    Note: View the original document on HAL open archive server: https://hal.science/hal-03671574
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