Norm convergence of powers of a Markov operator

Lundi 27 juin 2022, 11:00 à 12:00

Salle de séminaire M.0.1

Michael Lin

Université Ben Gourion du Néguev (Beer-Sheva, Israël)

Let $P(x,A)$ be a transition probability on $(X,\Sigma)$ and let $m$ be a probability on $\Sigma$ invariant for $P$, i.e. $m(A) =\int P(x,A)dm(x)$ for every $A \in \Sigma$. The Markov operator $Pf(x):= \int f(y)P(x,dy)$ is well-defined for $f$ bounded measurable; invariance of $m$ yields that $f=g$ a.e. (m) implies $Pf=Pg$ a.e. and $P$ is an operator on $L_\infty(m)$ and extends to an operator on $L_1(m)$. It is then a contraction in all $L_p(m)$, $1\le p \le \infty$. We assume that $P$ is ergodic modulo $m$, i.e. $Pf=f \in L_2(m)$ implies $f$ is a constant a.e.

We study the convergence of of the powers $P^{nd}$ for some $d$, in the strong operator topology of $L_2$. The convergence is connected to the deterministic $\sigma$-algebra of $P$ and its relation with the $\sigma$-algebras of invariant sets of the powers $P^k$.

Joint work with Guy Cohen.