Pesin's formula for isotropic Brownian flows

Pesin's formula is a relation between the entropy of a dynamical system and its positive Lyapunov exponents. This formula was first established by Pesin in the late 1970s for some deterministic dynamical systems acting on a compact Riemannian manifold. Later on the same formula was obtained in some other settings. For example, different authors have proved the formula for some random dynamical systems, or have relaxed the condition of state space compactness. Nevertheless, it has never been obtained for dynamical systems with invariant measure, which is infinite. The problem is that if invariant measure is infinite, then the notion of entropy becomes senseless. Invariant measure of isotropic Brownian flows is the Lebesgue measure on R^d, which is, clearly, infinite. Nevertheless, we define entropy for such a kind of flows using their translation invariance. For the definition we exploit ideas of Brin and Katok. Then we study the analogue of Pesin's formula for these flows using the defined entropy.