Foliated stochastic calculus: Harmonic measures

Catuogno, Pedro; Ledesma, Diego; Ruffino, Paulo

doi:10.1090/tran/6431

Foliated stochastic calculus: Harmonic measures
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by Pedro J. Catuogno, Diego S. Ledesma and Paulo R. Ruffino PDF

Trans. Amer. Math. Soc. 368 (2016), 563-579 Request permission

Abstract:

In this article we present an intrinsic construction of foliated Brownian motion (FoBM) via stochastic calculus adapted to a foliated Riemannian manifold $(M, \mathcal {F})$. The stochastic approach together with this proposed foliated vector calculus provide a natural method to work with (L. Garnett’s) harmonic measures in $M$. New results include, beside an explicit stochastic equation for the FoBM, a decomposition of the Laplacian of $M$ in terms of the foliated and the basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.

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