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Could anyone help me understand the definition on page 7 definition 2.25 here? I do not understand the notation $(P(a))(A)$ - what does this mean?

Also, is $P(a, A)$ a probability measure from the set of all Borel sets of $X$ to $[0,1]$?

I believe that $P$ is a function from $X \times B(X)\to [0,1]$?

Is this true?

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Going strictly by what Definition 2.25 in the link says, $P$ is a function $P : \mathcal{X} \to \mathcal{P}(\mathcal{X})$, where $\mathcal{P}(\mathcal{X})$ is the space of probability measures on $(\mathcal{X}, \mathscr{B}(\mathcal{X}))$ (presumably $\mathcal{X}$ is a topological space and $\mathscr{B}(\mathcal{X})$ is the $\sigma$-algebra of Borel subsets of $\mathcal{X}$, but I didn't read the notes to check whether that's true). That means that for each $a \in \mathcal{X}$, $P(a)$ is a probability measure on $(\mathcal{X}, \mathscr{B}(\mathcal{X}))$, so for each $A \in \mathscr{B}(\mathcal{X})$, $(P(a))(A)$ is a number between $0$ and $1$.

You can think of $P$ as a "random probability measure" if you equip $\mathcal{P}(\mathcal{X})$ with a $\sigma$-algebra that makes $P$ a measurable function. Usually this is done by endowing $\mathcal{P}(\mathcal{X})$ with the $\sigma$-algebra $\mathcal{H}$ which is the smallest $\sigma$-algebra such that for each $B \in \mathscr{B}(\mathcal{X})$, the map $Q \mapsto Q(B)$ from $\mathcal{P}(\mathcal{X})$ to $[0, 1]$ is measurable (i.e., $\mathcal{H}$ is generated by all such maps). This way, $P : \mathcal{X} \to \mathcal{P}(\mathcal{X})$ is a measurable function, hence a "random element" of $\mathcal{P}(\mathcal{X})$.

The random probability measure approach is one way of defining the transition probability $P$. Another approach is using so-called transition kernels (AKA Markov kernels, etc.).

If $(\mathcal{X}, \mathcal{F})$ and $(\mathcal{Y}, \mathcal{G})$ are two measure spaces, then a transition kernel from $(\mathcal{X}, \mathcal{F})$ to $(\mathcal{Y}, \mathcal{G})$ is a function $Q : \mathcal{X} \times \mathcal{G} \to [0, 1]$ such that

  1. for each $x \in \mathcal{X}$, the map $Q(x, \cdot) : \mathcal{G} \to [0, 1]$ is a probability measure on $(\mathcal{Y}, \mathcal{G})$, and
  2. for each $G \in \mathcal{G}$, the map $Q(\cdot, G) : \mathcal{X} \to [0, 1]$ is $\mathcal{F}$-measurable.

Given a transition kernel $Q$ from $(\mathcal{X}, \mathcal{F})$ to $(\mathcal{Y}, \mathcal{G})$, we can define a random measure $P : \mathcal{X} \to \mathcal{P}(\mathcal{Y})$, where $\mathcal{P}(\mathcal{Y})$ stands for the space of probability measures on $(\mathcal{Y}, \mathcal{G})$, by defining $$ (P(a)(A)) = Q(a, A) $$ for each $a \in \mathcal{X}$ and $A \in \mathcal{G}$. Conversely, given a random measure $P : \mathcal{X} \to \mathcal{P}(\mathcal{Y})$, we can define a transition kernel $Q$ from $(\mathcal{X}, \mathcal{F})$ to $(\mathcal{Y}, \mathcal{G})$ by the same relation. Thus, there is a one-to-one correspondence between random probability measures and transition kernels.

Because of this, it is common to use notation like $P(a, A)$ when $P$ is a random probability measure to stand for $(P(a))(A)$; the notation $P(a, A)$ is easier to read, and you're not really doing anything wrong by treating the random probability measure as a transition kernel.

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In homogeneous markov chain the system dynamics do not change over time, I think it would be just the expression of the transition kernel which is fixed with time.

Maybe the typing in this textbook would be more expressive:
enter image description here

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  • $\begingroup$ Thanks for your answer, actually I am interested in understanding the notations and definition from the Hairer notes only, for my context. $\endgroup$ – riemann May 24 at 9:45

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