I am reading the book Brownian motion by Peter Morters and Yuval Peres. The following (Markov Transition Kernel) is defined in section 2.3 (Markov processes derived from Brownian motion). It is later used as a notation in proof of Levy's theorem about running the maximum of linear Brownian motion. I don't get the intuition for defining this quantity, also exactly what it means:
Definition 2.30. A function $p:[0, \infty) \times \mathbb{R}^{d} \times \mathfrak{B} \rightarrow \mathbb{R}$, where $\mathfrak{B}$ is the Borel $\sigma$-algebra in $\mathbb{R}^{d}$, is a Markov transition kernel provided
(1) $p(\cdot, \cdot, A)$ is measurable as a function of $(t, x)$, for each $A \in \mathfrak{B}$;
(2) $p(t, x, \cdot)$ is a Borel probability measure on $\mathbb{R}^{d}$ for all $t \geqslant 0$ and $x \in \mathbb{R}^{d}$, when integrating a function $f$ with respect to this measure we write
$$
\int f(y) p(t, x, d y)
$$
(3) for all $A \in \mathfrak{B}, x \in \mathbb{R}^{d}$ and $t, s>0$,
$$
p(t+s, x, A)=\int_{\mathbb{R}^{d}} p(t, y, A) p(s, x, d y) .
$$
Any help is highly appreciated. Thanks!
Best Answer
An intuitive interpretation: Suppose that this describes the motion of a particle the trajectory is given by $X(t)$.
If $X(t_0)=x$ then the conditional probability of finding it in some Borel set $A$ at time $t_0+t$ ($t>0$) is given by $$P(X(t_0+t)\in A| X(t_0)=x) = p(t,x,A).$$
Loss of memory: If $X(t_0)=x$ and $X(t_0+s)=y$ ($s>0$) then the conditional probability of finding the particle in a Borel set $A$ at time $t_0+s+t$ ($t>0$) is $$P(X(t_0+t)\in A| X(t_0+s)=y, X(t_0)=x) = p(t,y,A).$$ Thus there is no memory of what happened before the time $t_0+s$. The value of $x$ is irrelevant.
The latter then translates into the integral condition of the kernel when integrating over the intermediate positions of $y$ at time $t_0+s$. (like Bayes' formula in integral version): $$ P(X(t_0+s+t)\in A | X(t_0)=x) =\\ \int P(X(t_0+s+t)\in A| X(t_0+s)=y, X(t_0)=x) P(X(t_0+s)\in dy| X(t_0)=x)$$ or simply $$ p(s+t,x,A) =\int p(t,y,A) p(s,x,dy).$$