Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds   In this paper we introduce three Markovian couplings of Brownian motions on
smooth Riemannian manifolds without boundary which sit at the crossroad of two
concepts. The first concept is the one of shy coupling put forward in
\cite{Burdzy-Benjamini} and the second concept is the lower bound on the Ricci
curvature and the connection with couplings made in \cite{ReSt}.
  The first construction is the shy coupling, the second one is a
fixed-distance coupling and the third is a coupling in which the distance
between the processes is a deterministic exponential function of time.
  The result proved here is that an arbitrary Riemannian manifold satisfying
some technical conditions supports shy couplings. If in addition, the Ricci
curvature is non-negative, there exist fixed-distance couplings. Furthermore,
if the Ricci curvature is bounded below by a positive constant, then there
exists a coupling of Brownian motions for which the distance between the
processes is a decreasing exponential function of time. The constructions use
the intrinsic geometry, and relies on an extension of the notion of frames
which plays an important role for even dimensional manifolds.
  In fact, we provide a wider class of couplings in which the distance function
is deterministic in Theorem \ref{t:100} and Corollary~\ref{Cor:9}.
  As an application of the fixed-distance coupling we derive a maximum
principle for the gradient of harmonic functions on manifolds with non-negative
Ricci curvature.
  As far as we are aware of, these constructions are new, though the existence
of shy couplings on manifolds is suggested by Kendall in \cite{Kendall}.

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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci