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.

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 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.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

2017 ◽  
Vol 59 (3) ◽  
pp. 743-751
Author(s):  
SHOUWEN FANG ◽  
FEI YANG ◽  
PENG ZHU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Normalized Ricci Flow ◽  
Geometric Operator

AbstractLet (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

The Palais–Smale conditions for the Yang–Mills functional

1988 ◽  
Vol 108 (3-4) ◽  
pp. 189-200
Author(s):  
D. R. Wilkins
Keyword(s):  
Riemannian Manifold ◽  
Principal Bundle ◽  
Compact Riemannian Manifold ◽  
Hilbert Manifold ◽  
Yang Mills ◽  
Palais Smale Condition ◽  
Dimension 2

SynopsisWe consider the Yang–Mills functional denned on connections on a principal bundle over a compact Riemannian manifold of dimension 2 or 3. It is shown that if we consider the Yang–Mills functional as being defined on an appropriate Hilbert manifold of orbits of connections under the action of the group of principal bundle automorphisms, then the functional satisfies the Palais–Smale condition.

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