scholarly journals Harmonic measures in embedded foliated manifolds

2017 ◽  
Vol 17 (04) ◽  
pp. 1750026
Author(s):  
Pedro J. Catuogno ◽  
Diego S. Ledesma ◽  
Paulo R. Ruffino
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Compact Riemannian Manifold ◽  
Harmonic Measures ◽  
Foliated Manifolds ◽  
Stratonovich Differential

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold. We construct explicitly a Stratonovich differential equation for the foliated Brownian motion. We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.

scholarly journals Curvature, geodesics and the Brownian motion on a Riemannian manifold I—Recurrence properties

1982 ◽  
Vol 87 ◽  
pp. 101-114 ◽  
Author(s):  
Kanji Ichihara
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Locally Compact

LetMbe ann-dimensional, complete, connected and locally compact Riemannian manifold andgbe its metric. Denote byΔMthe Laplacian onM.

scholarly journals Some ergodic properties of commuting diffeomorphisms

1993 ◽  
Vol 13 (1) ◽  
pp. 73-100 ◽  
Author(s):  
Huyi Hu
Keyword(s):  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Tangent Space ◽  
Compact Riemannian Manifold ◽  
Ergodic Properties

AbstractFor a smooth ℤ2-action on a C∞ compact Riemannian manifold M, we discuss its ergodic properties which include the decomposition of the tangent space of M into subspaces related to Lyapunov exponents, the existence of Lyapunov charts, and the subadditivity of entropies.

scholarly journals The property of convex carrying simplices for competitive maps

2018 ◽  
Vol 40 (5) ◽  
pp. 1335-1350 ◽  
Author(s):  
JANUSZ MIERCZYŃSKI
Keyword(s):  
Lyapunov Exponents ◽  
Invariant Measures

For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In this paper it is shown that its convexity implies that it is a $C^{1}$ submanifold-with-corners, neatly embedded in the non-negative orthant. The proof uses the characterization of neat embedding in terms of inequalities between Lyapunov exponents for ergodic invariant measures supported on the boundary of the carrying simplex.

scholarly journals THE SMOLYANOV SURFACE MEASURE ON TRAJECTORIES IN A RIEMANNIAN MANIFOLD

2004 ◽  
Vol 07 (03) ◽  
pp. 461-471 ◽  
Author(s):  
NADEZDA A. SIDOROVA
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Direct Proof ◽  
Weak Limit ◽  
Natural Object ◽  
Surface Measure ◽  
Definition Of ◽  
Special Case ◽  
Restrictive Definition

It has been shown in Refs. 2–6 that two natural definitions of surface measures, on the space of continuous paths in a compact Riemannian manifold embedded into ℝn, introduced in the paper by Smolyanov1 are equivalent; this means that there exists a natural object — the surface measure, which we call the Smolyanov surface measure. Moreover, it has been shown2–6 that this surface measure is equivalent to the Wiener measure and the corresponding density has been found. But the known proof of the equivalence of the two definitions of the surface measure is rather nonexplicit; in fact the densities of the measures corresponding to the two different definitions were found independently and only a posteriori it was discovered that those densities coincided. Our aim is to give a direct proof of this fact. We introduce a more restrictive definition of the surface measure as the weak limit of a standard Brownian motion in ℝn conditioned to be in the tubular ε-neighborhood of the manifold at times 0=t0<t1<⋯<tn-1<tn= 1 as both ε and the diameter of the partition tend to zero. Letting ε and then the diameter of the partition tend to zero and vice versa, we arrive at the two definitions above. We prove the existence of the Smolyanov surface measure using our definition, show that this measure is equivalent to the law of a Brownian motion on the manifold, and compute the corresponding density in terms of the curvature of the manifold. As a special case of this, we again obtain the results of Refs. 2–6.

scholarly journals On an Anti-Torqued Vector Field on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2201
Author(s):  
Sharief Deshmukh ◽  
Ibrahim Al-Dayel ◽  
Devaraja Mallesha Naik
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Dual Vector ◽  
Definition Of ◽  
Simply Connected

A torqued vector field ξ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative.

scholarly journals Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold

2019 ◽  
Vol 178 (1) ◽  
pp. 75-116
Author(s):  
Bart van Ginkel ◽  
Frank Redig
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Heat Equation ◽  
Random Walks ◽  
Hydrodynamic Limit ◽  
Compact Riemannian Manifold ◽  
Exclusion Process ◽  
Density Field ◽  
Random Grids ◽  
Empirical Density

Abstract We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it converges to the solution of the heat equation on the manifold.

scholarly journals Curvature, geodesics and the Brownian motion on a Riemannian manifold II—Explosion properties

1982 ◽  
Vol 87 ◽  
pp. 115-125 ◽  
Author(s):  
Kanji Ichihara
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Compact Riemannian Manifold

Let M be an n-dimensional, complete, connected and non compact Riemannian manifold and g be its metric. ΔM denotes the Laplacian on M.

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