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

TL;DR

This paper investigates how a specific perturbation based on Gaussian curvature influences foliated Brownian motion, revealing a linear behavior linked to the Euler characteristic of the leaves as the perturbation diminishes.

Contribution

It introduces an averaging principle for foliated diffusions with curvature-dependent perturbations, connecting topology and stochastic dynamics.

Findings

Transversal component approaches a linear function proportional to Euler characteristic.

Rate of convergence of the perturbation effect is estimated.

Behavior is characterized after rescaling time by 1/epsilon.

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 $\R^3$. 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.