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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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