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Application of an averaging principle on foliated diffusions: topology of the leaves. (English) Zbl 1327.60159
Summary: 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.

MSC:
 60J60 Diffusion processes 60J65 Brownian motion 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 58J65 Diffusion processes and stochastic analysis on manifolds 58J37 Perturbations of PDEs on manifolds; asymptotics
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