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 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 On Weak Limiting Distributions for Random Walks on a Spider

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2000
Author(s):  
Youngsoo Seol
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Classical Case ◽  
Weak Limit ◽  
Skew Brownian Motion ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Dimensional Distribution ◽  
Functional Central Limit ◽  
Special Case

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider and to define Brownian spider as the resulting weak limit. In special case, random walks on a spider can be characterized as skew random walks. It is well known for skew Brownian motion as the resulting weak limit of skew random walks. We first will study the tightness and then it will be shown for the convergence of finite dimensional distribution for random walks on a spider.

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.

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 Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

Sign in / Sign up

Export Citation Format

Share Document