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Foliated stochastic calculus: Harmonic measures


Authors: Pedro J. Catuogno, Diego S. Ledesma and Paulo R. Ruffino
Journal: Trans. Amer. Math. Soc. 368 (2016), 563-579
MSC (2010): Primary 58J65, 53C12; Secondary 60H30, 60J60
DOI: https://doi.org/10.1090/tran/6431
Published electronically: April 24, 2015
MathSciNet review: 3413874
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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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Pedro J. Catuogno
Affiliation: Departamento de Matemática, Universidade Estadual de Campinas, 13.081-970 - Campinas - SP, Brazil
Email: pedrojc@ime.unicamp.br

Diego S. Ledesma
Affiliation: Departamento de Matemática, Universidade Estadual de Campinas, 13.081-970 - Campinas - SP, Brazil
Email: dledesma@ime.unicamp.br

Paulo R. Ruffino
Affiliation: Departamento de Matemática, Universidade Estadual de Campinas, 13.081-970 - Campinas - SP, Brazil
Email: ruffino@ime.unicamp.br

DOI: https://doi.org/10.1090/tran/6431
Keywords: Foliation, diffusion process, stochastic calculus
Received by editor(s): August 1, 2012
Received by editor(s) in revised form: November 22, 2013
Published electronically: April 24, 2015
Additional Notes: The first author’s research was partially supported by FAPESP 11/50151-0 and 12/18780-0.
Article copyright: © Copyright 2015 American Mathematical Society