Lecture 4. Markovian semigroups

Let $(X, \mathcal{B})$ be a measurable space. We say that $(X, \mathcal{B})$ is a good measurable space if there is a countable family generating $\mathcal{B}$ and if every finite measure $\gamma$ on $(X \times X, \mathcal{B} \otimes \mathcal{B})$ can be decomposed as

$\gamma (dx dy)=k(x,dy) \gamma_1 (dx)$

where $\gamma_1$ is the projection of $\gamma$ on the first coordinate and $k$ is a kernel, i.e $k(x,\cdot)$ is a finite measure on $(X, \mathcal{B})$ and $x \to k(x,A)$ is measurable for every $A \in \mathcal{B}$ .

For instance, if $X$ is a Polish space equipped with its Borel $\sigma$ -field, then it is a good measurable space.

Throughout the lecture, we will consider $(X, \mathcal{B}, \mu)$ to be a good measurable space equipped with a $\sigma$ -finite measure $\mu$ .

Definition: Let $(P_t)_{t \ge 0}$ be a strongly continuous self-adjoint contraction semigroup on $L^2(X,\mu)$ . The semigroup $(P_t)_{t \ge 0}$ is called Markovian if and only if for every $f \in L^2(X,\mu)$ and $t \ge 0$ :

1) $f \ge 0, \text{ a.e } \implies P_t f \ge 0$ a.e.

2) $f \le 1, \text{ a.e } \implies P_t f \le 1$ , a.e..

We note that if $(P_t)_{t \ge 0}$ is Markovian, then for every $f \in L^2(X,\mu) \cap L^\infty(X,\mu)$ ,
$\| P_t f \|_{L^\infty(X,\mu)} \le \| f \|_{L^\infty(X,\mu)}.$
As a consequence $(P_t)_{t \ge 0}$ can be extended to a contraction semigroup defined on all of $L^\infty(X,\mu)$ .

Definition: A transition function $\{p_t,t \geq 0 \}$ on $X$ is a family of kernels
$p_t : X \times \mathcal{B}\rightarrow [0,1]$
such that:
1) For $t \geq 0$ and $x \in X$ , $p_t (x,\cdot)$ is a finite measure on $X$ ;
2) For $t \geq 0$ and $A \in \mathcal{B}$ the application
$x \rightarrow p_t (x,A)$ is measurable;
3) For $s,t \geq 0$ , a.e. $x \in X$ and $A\in \mathcal{B}$ ,
$p_{t+s} (x,A)=\int_{X} p_t(y,A) p_s (x,dy).$

The relation 3) is often called the Chapman-Kolmogorov relation

Theorem A: Let $(P_t)_{t \ge 0}$ be a strongly continuous self-adjoint contraction Markovian semigroup on $L^2(X,\mu)$ . There exists a transition function $\{p_t,t \geq 0 \}$ on $X$ such that for every $f \in L^\infty(X,\mu)$ and a.e. $x \in X$
$P_tf (x)=\int_X f(y) p_t(x,dy), \quad t > 0.$
This transition function is called the heat kernel measure associated to $(P_t)_{t \ge 0}$ .

The proof relies on the following lemma sometimes called the bi-measure theorem. A set function $\nu: \mathcal{B} \otimes \mathcal{B} \to [0,+\infty)$ is called a bi-measure, if for every $A \in \mathcal{B}$ , $\nu (A, \cdot)$ and $\nu(\cdot,A)$ are measures.

Lemma: If $\nu: \mathcal{B} \otimes \mathcal{B} \to [0,+\infty)$ is a bi-measure, then there exists a measure $\gamma$ on $\mathcal{B} \otimes \mathcal{B}$ such that for every $A,B \in \mathcal{B}$ ,
$\gamma (A \times B)=\nu(A,B).$

Proof of Theorem A: We assume that $\mu$ is finite and let as an exercise the extension to $\sigma$ -finite measures. For $t>0$ , we consider the set function
$\nu_t(A,B)=\int_X 1_A P_t 1_B d\mu.$
Since $P_t$ is supposed to be Markovian, it is a bi-measure. From the bi-measure theorem, there exists a measure $\gamma_t$ on $\mathcal{B} \otimes \mathcal{B}$ such that for every $A,B \in \mathcal{B}$ ,
$\gamma_t (A \times B)=\nu_t(A,B)=\int_X 1_A P_t 1_B d\mu.$
The projection of $\gamma_t$ on the first coordinate is $\mu$ , thus from the measure decomposition theorem, $\gamma_t$ can be decomposed as
$\gamma_t (dx dy)=p_t(x,dy) \mu (dx)$
for some kernel $p_t$ . One has then for every $A,B \in \mathcal{B}$
$\int_X 1_A P_t 1_B d\mu=\int_A \int_B p_t(x,dy) \mu (dx),$
from which it follows that for every $f \in L^\infty(X,\mu)$ , and a.e. $x \in X$
$P_tf (x)=\int_X f(y) p_t(x,dy).$
The relation
$p_{t+s} (x,A)=\int_{X} p_t(y,A) p_s (x,dy)$
follows from the semigroup property. $\square$

Exercise: Prove Theorem A if $\mu$ is $\sigma$ -finite.

Exercise: Show that for every non-negative measurable function $F: X \times X \to \mathbb{R}$ , $\int_X \int_X F(x,y) p_t(x,dy) d\mu(x)=\int_X \int_X F(x,y) p_t(y,dx) d\mu(y).$

2 Responses to Lecture 4. Markovian semigroups

Dennis Sullivan says:

February 11, 2019 at 2:07 am

hi fabrice, this is dennis sullivan writing.

do you know of or suspect a structure theory for strong markovian semigroups on say the d torus, with finite stationary measures where the paths have parametrizations of finite energy and are therefore lipschitz AE

thanks dennis sullivan

- Fabrice Baudoin says:
  
  February 11, 2019 at 3:20 pm
  
  Hi Dennis, Yes there are some nice dynamics on paths of finite energy. In the preprint https://arxiv.org/pdf/1605.02192.pdf , M. Hairer describes lines of research along those lines. For the torus, the dynamic is simpler since the Christofell symbols vanish.