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General framework Analysis of the spectrum

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31 pages
General framework Analysis of the spectrum Convergence to stationary measure Sketches of proof Semiclassical analysis of a random walk on a manifold L. Michel (joint work with G. Lebeau) Laboratoire J.-A. Dieudonne Universite de Nice April 14, 2008 L. Michel (joint work with G. Lebeau) Semiclassical analysis of a random walk on a manifold

  • ?2 ≤

  • natural random

  • basic properties

  • reference operator

  • associated normalized

  • laplace-beltrami operator

  • operator ?∆g


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L. Michel (joint work with G. Lebeau)
LaboratoireJ.-A.Dieudonne´ Universite´deNice
April 14, 2008
Semiclassical analysis of a random walk on a manifold
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(x) =|B(x1,h)|ZB(x,h)f(y)dgy (Thf)
Let (M,gbe a smooth, compact, connected Riemannian manifold) of dimension d, equipped with its canonical volume formdgx. We denote dg(x,y) the Riemannian distance onM×M. forxMandh>0,B(x,h) ={y,dg(x,y)h} |B(x,h)|=RB(x,h)dgy For any givenh>0, letThbe the operator acting on continuous functions onM
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We denote byKhthe kernel ofTh, which is given by
Kh(x,y)dgy=1|{dBg((xx,,yh))|h}dgy
for anyxM,Kh(x,y)dgyis a probability measure onM (hence,Khis a Markov kernel) Khis associated to the following natural random walk (Xn) on M:if the walk is atx, then it moves to a pointyB(x,h) with a probability given byKh(x,y)dgy. ForfC0(M), Th(f)(Xn) =E(f(Xn+1)|Xn). Or equivalently, forA,B measurable,
P(Xn+1AandXnB) =E(Th(1A)(Xn)1B(Xn)).
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