General framework Analysis of the spectrum

Dieudonne Universite

Description

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

Sujets

• Université
• Laplace?Beltrami operator

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

Extrait

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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(x 1 ,h)| Z B(x,h) f(y)d g y (T h f)

Let (M,gbe a smooth, compact, connected Riemannian manifold) of dimension d, equipped with its canonical volume formd g x. We denote d g (x,y) the Riemannian distance onM×M. forx∈Mandh>0,B(x,h) ={y,d g (x,y)≤h} |B(x,h)|= R B(x,h) d g y For any givenh>0, letT h be the operator acting on continuous functions onM

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We denote byK h the kernel ofT h , which is given by

K h (x,y)d g y= 1 | {d B g ( (x x , , y h )≤ )| h} d g y

for anyx∈M,K h (x,y)d g yis a probability measure onM (hence,K h is a Markov kernel) K h is associated to the following natural random walk (X n ) on M:if the walk is atx, then it moves to a pointy∈B(x,h) with a probability given byK h (x,y)d g y. Forf∈C 0 (M), T h (f)(X n ) =E(f(X n+1 )|X n ). Or equivalently, forA,B measurable,

P(X n+1 ∈AandX n ∈B) =E(T h (1 A )(X n )1 B (X n )).