Department of Mathematics

Indian Institute of Science

Bangalore 560 012

 

SEMINAR

 

Speaker

:

 Prof Ewa Damek
Affiliation : Wroclaw University

Subject Area

:

Mathematics

 

Venue

:

Department of Mathematics, Lecture Hall I

 

Time

:

4.00 p.m.-5.00 p.m.

 

Date  

:

September14, 2012 (Friday)

Title

:

"From Poisson kernels to stationary measures for random recursions"
Abstract

:

Let S be a group which is a semi-direct product of R^n and R^d, R^d acting of R^n in a appropriate way (that will be specified during the talk). Given a left-invariant second order elliptic operator L on S (under suitable assumptions) there is smooth, positive integrable function on R^n called a Poisson kernel that reproduces bounded L harmonic functions on S. It turns out that P dx is the stationary measure for a random recursion X_{n+1}=f_{n+1}(X_n) where f_1, f_2,... are independent equally distributed random transfor -mations of R^n, and X_n are defined recursively, X_0 being a constant vector. The law of f_1 is closely related to the heat kernel for L. P dx is the law of the limit of X_n. The situation can be generalized to transformations f having the law with no relation to a diffe -rential operator giving rise to a family of affine stochastic recursions who's stationary measures are of interest. Among them are the classical ones considered by Kesten, Vervaat, Grincevicius, Goldie and more recent ones considered by Alsmeyer - Mentemeier, Guivarc'h - Le Page, Guivarc'h - Buraczewski and myself. Asymptotics at infinity of stationary measures will be described.