Representation of the $n$-th power of an operator given by a transition kernel

Question

If $P(x,A)$ is a transitional kernel on a space $X$, it can act as an operator on the space of bounded measurable functions by

$$(Pf)(x)=\int f(y) P(x,dy).$$

Could anyone tell me what the $n$-th power $P^n f$ would look like?

Also, could anyone tell me where the factor $2$ in Theorem 1.10 here comes from?

Thanks.

Answer 1

You have already defined the function $P^1 f$ by

$$(P^1 f)(x)=(Pf)(x)=\int f(y)P(x,dy)$$

for all $x\in X$. For $n\ge2$, you can now use recursion to define

$$P^n f(x):= (P(P^{n-1}f))(x)$$

for all $x\in X$. This means that

\begin{align} P^2f(x) &= \int (Pf)(y)P(x,dy) \\ &=\int \left(\int f(z) P(y,dz) \right) P(x,dy)\\ &=\int P(x,dy) \int P(y,dz) f(z), \end{align}

then

\begin{align} P^3f(x) &= \int (P^2f)(y)P(x,dy) \\ &=\int P(x,dy) \int P(y,dz) \int P(z,dw) f(w), \end{align}

and so on.

Your second question is entirely unrelated and you should actually open a new question for it. Anyway, the step you probably mean can be explained by

$$ \| P^N\varphi-\varphi\|\le \| P^N\varphi\|+\|\varphi\|\le 2\|\varphi\|.$$