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

TL;DR

This paper derives gradient estimates and a Liouville property for the kinetic Brownian motion in the plane, focusing on large time behavior using Malliavin calculus and integration by parts.

Contribution

It introduces an explicit Malliavin calculus approach to analyze the large time behavior of the kinetic Brownian motion in the plane, including gradient bounds and harmonic function properties.

Findings

Gradient estimates for the semi-group

Liouville property for the generator

Explicit Malliavin calculus computation

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{\'e} 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.