Application of an averaging principle on foliated diffusions: topology of the leaves

Abstract

We consider an $\epsilon K$ transversal perturbing vector field in a foliated Brownian motion defined in a foliated tubular neighbourhood of an embedded compact submanifold in ${\mathbb{R}}^3$. We study the effective behaviour of the system under this $\epsilon$ perturbation. If the perturbing vector field $K$ is proportional to the Gaussian curvature at the corresponding leaf, we have that the transversal component, after rescaling the time by $t/\epsilon$, approaches a linear increasing behaviour proportional to the Euler characteristic of $M$, as $\epsilon$ goes to zero. An estimate of the rate of convergence is presented.