Work

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

Year: 2026

Type: article

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 $ L^2([0,T], \mathbb R) $ and an independent sequence of $ {\mathscr 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.

Source: Communications on Pure &amp Applied Analysis

DOI: 10.3934/cpaa.2026100

Authors: Marc Arnaudon, Magalie Bénéfice, Michel Bonnefont, Delphine Féral

Institutions: Centre National de la Recherche Scientifique, Institut Polytechnique de Bordeaux, Institut de Mathématiques de Bordeaux, Institut Élie Cartan de Lorraine, Université de Lorraine

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Agence Nationale de la Recherche

OpenAlex ID: W7171982381

FWCI:

Cites: 31

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Related to: 10