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 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.

The Proof of the Feynman–Kac Formula for Heat Equation on a Compact Riemannian Manifold

2003 ◽  
Vol 06 (02) ◽  
pp. 311-320 ◽  
Author(s):  
Oleg O. Obrezkov
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Compact Riemannian Manifold ◽  
Type Formula ◽  
Chernoff Theorem ◽  
Common Lines ◽  
Full Proof

A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.

Spherical Mean and the Fundamental Group

1991 ◽  
Vol 34 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Random Walks ◽  
Isoperimetric Inequality ◽  
Lower Bounds ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Spherical Means ◽  
Universal Covering Space ◽  
Spherical Mean

AbstractWe investigate some properties of spherical means on the universal covering space of a compact Riemannian manifold. If the fundamental group is amenable then the greatest lower bounds of the spectrum of spherical Laplacians are equal to zero. If the fundamental group is nontransient so are geodesic random walks. We also give an isoperimetric inequality for spherical means.

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 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.

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