Degenerate semigroups and stochastic flows of mappings in foliated manifolds

2013/06/01 by Paulo Henrique P da Costa, da Costa, Paulo Henrique P, Paulo R. Ruffino +1
Mathematics · #57C12 #60H10 #60H30 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:57C12 #msc:60H10 #msc:60H30

paper · pdf · doi:10.48550/arxiv.1306.0123

arxiv created 2013/06/01 · arxiv updated 2013/06/04

Abstract

Let (M, F) be a compact Riemannian foliated manifold. We consider a family of compatible Feller semigroups in C(Mⁿ) associated to laws of the n-point motion. Under some assumptions (Le Jan and Raimond, \citeLe Jan-Raimond) there exists a stochastic flow of measurable mappings in M. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated.

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