Mathematical Research Letters

Volume 26 (2019)

Number 2

Lyapunov exponents of the Brownian motion on a Kähler manifold

Pages: 501 – 536



Jeremy Daniel (Max Planck Institute for Mathematics, Bonn, Germany)

Bertrand Deroin (Département mathématiques, Université Cergy-Pontoise, France)


If $E$ is a flat bundle of rank $r$ over a Kähler manifold $X$, we define the Lyapunov spectrum of $E$: a set of $r$ numbers controlling the growth of flat sections of $E$, along Brownian trajectories. We show how to compute these numbers, by using harmonic measures on the foliated space $\mathbb{P}(E)$. Then, in the case where $X$ is compact, we prove a general inequality relating the Lyapunov exponents and the degrees of holomorphic subbundles of $E$ and we discuss the equality case.

Received 18 January 2018

Accepted 1 May 2018

Published 12 August 2019