Skip to content
DraconiticResearch
Search Sources API
182,469 studies31 sourcesUpdated 31 min ago
Peer-reviewedCC BY 4.0Communications on Pure and Applied Analysis

Large time behaviour for the semigroup of the kinetic Brownian motion in the plane

  • Marc Arnaudon
  • Magalie Bénéfice
  • Michel Bonnefont
  • Delphine Féral
Open at sourcePDFDOICite

Abstract

We establish an integration by parts formula for the semigroup in time T>0 of the kinetic Brownian motion in the Euclidean plane together with its velocity in the circle. The stochastic differential equation of our kinetic Brownian motion is driven here by one real-valued Brownian motion constructed from an orthonormal basis of L2([0,T],ℝ) and an independent sequence of 𝒩(0,1) random variables. Our method is based on an explicit computation of a Malliavin dual in the Gaussian space. We are mainly interested in large time T. From our integration by parts, we obtain gradient estimates including a reverse Poincaré inequality for the semigroup. As a direct consequence, we also obtain a Liouville property for the generator of the kinetic Brownian motion and its speed: all bounded harmonic functions are constant.

Related

  1. Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
    Yan V Fyodorov, Pierre Le Doussal, Alexander Ossipov
    Journal of Mathematical PhysicsPeer-reviewedCC BY 4.0Full textMathematical physicsHAL
  2. p-Brownian motion and the p-Laplacian
    Viorel Barbu, Marco Rehmeier, Michael Röckner
    The Annals of ProbabilityPeer-reviewedLicence not statedFull text5 citedGeneral mathematicsOpenAlex
  3. PERSISTENCE OF THE WIENER SAUSAGE: SAMPLING STABILITY AND LIMIT THEOREMS FOR BETTI-CURVE FUNCTIONALS OF DRIFTED PLANAR BROWNIAN MOTION
    Tristan Guillaume
    HALPreprintPublisher termsFull textProbabilityHAL
  4. Parametric estimation for stochastic wave equations driven by an infinite dimensional fractional Brownian motion
    B. L. S. Prakasa Rao
    arXivPreprintCC BY 4.0Full textProbabilityarXiv
  5. Fractional Brownian motion with multivariate time: persistence exponents
    George Molchan
    arXivPreprintarXiv licenceFull textProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@article{arnaudon2026large,
  title = {{Large time behaviour for the semigroup of the kinetic Brownian motion in the plane}},
  author = {Marc Arnaudon and Magalie Bénéfice and Michel Bonnefont and Delphine Féral},
  journal = {Communications on Pure and Applied Analysis},
  year = {2026},
  month = jul,
  doi = {10.3934/cpaa.2026100},
  url = {https://hal.science/hal-05556406v2},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Large time behaviour for the semigroup of the kinetic Brownian motion in the plane
AU  - Marc Arnaudon
AU  - Magalie Bénéfice
AU  - Michel Bonnefont
AU  - Delphine Féral
PY  - 2026
DA  - 2026/07/31
JO  - Communications on Pure and Applied Analysis
DO  - 10.3934/cpaa.2026100
UR  - https://hal.science/hal-05556406v2
AB  - We establish an integration by parts formula for the semigroup in time T>0 of the kinetic Brownian motion in the Euclidean plane together with its velocity in the circle. The stochastic differential equation of our kinetic Brownian motion is driven here by one real-valued Brownian motion constructed from an orthonormal basis of L^2([0,T],ℝ) and an independent sequence of 𝒩(0,1) random variables. Our method is based on an explicit computation of a Malliavin dual in the Gaussian space. We are mainly interested in large time T. From our integration by parts, we obtain gradient estimates including a reverse Poincaré inequality for the semigroup. As a direct consequence, we also obtain a Liouville property for the generator of the kinetic Brownian motion and its speed: all bounded harmonic functions are constant.
SN  - 1534-0392
SN  - 1553-5258
ER  -