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

Magalie Bénéfice, Michel Bonnefont, Marc Arnaudon, Delphine Féral

Abstract

We establish an integration by parts formula for the semi-group in time $T > 0$ of the kinetic Brownian motion in the Euclidean plane together with its speed in the circle. The stochastic differential equation of our kinetic Brownian motion is driven here by one real-valued Brownian motion constructed with 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 Poincar{é} inequality for the semi-group. 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.

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

Abstract

We establish an integration by parts formula for the semi-group in time of the kinetic Brownian motion in the Euclidean plane together with its speed in the circle. The stochastic differential equation of our kinetic Brownian motion is driven here by one real-valued Brownian motion constructed with an orthonormal basis of and an independent sequence of 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 . From our integration by parts, we obtain gradient estimates including a reverse Poincar{é} inequality for the semi-group. 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.
Paper Structure (17 sections, 18 theorems, 221 equations)

This paper contains 17 sections, 18 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. The context of our study
  3. Organisation and main results
  4. Presentation of the model and Malliavin calculus
  5. The model on $\mathbb S \times \mathbb C$ and its lift on $\mathbb R\times \mathbb C$
  6. The tangent flow
  7. Malliavin derivative and integration by parts formula
  8. The infinitesimal control problem and the Malliavin matrix
  9. Malliavin calculus in a basis and computation of the dual
  10. Stochastic oscillating integrals
  11. Integrability and convergence of the dual
  12. Asymptotic behaviour of the Malliavin matrix
  13. Moments convergence of the renormalized matrix $\tilde{\mathcal{C}}(T)^{-1}$
  14. Proofs of Lemmas \ref{['L3']}, \ref{['L1']} and \ref{['LY']}
  15. Application to the dual
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $f : \mathbb R\times \mathbb C \to \mathbb R$ be a bounded measurable function. Denote by $P_T$ the semigroup of $X_T$ and $\delta$ the Malliavin duality operator. If $h$ is in the domain of $\delta$ and satisfies eq:T+D=0, then we have

Theorems & Definitions (40)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.4
  • ...and 30 more