I have a problem showing that "A $d$-dimensional Brownian family is a Markov family." by Karatzas/Shreve in the book "Brownian Motion and Stochastic Calculus" (cf. p. 75 Theorem 2.5.12). Let me first recall some definitions in the book:
5.1 Definition. Let $d$ be a positive integer and $\mu$ a probability measure on $\left(\mathbb{R}^{d},\mathscr{B}\left(\mathbb{R}^{d}\right)\right)$ Let $B=\left\{ B_{t},\mathscr{F}_{t}:0\le t<\infty\right\}$ be a continuous, adapted process with values in $\mathbb{R}^{d}$, defined on some probability space $\left(\Omega,\mathscr{F},P\right)$. This process is called a $d$-dimensional Brownian motion with initial distribution $\mu$, if
• (i) $P\left(B_{0}\in A\right)=\mu\left(A\right)$, for all $A\in\mathscr{B}\left(\mathbb{R}^{d}\right)$
• (ii) for $0\le s<t<\infty$, the increment $B_{t}-B_{s}$ is independent of $\mathscr{F}_{s}$ and is normally distributed with mean zero and covariance matrix equal to $\left(t-s\right)I_{d}$, where $I_{d}$ is the $\left(d\times d\right)$ identity matrix.
If $\mu$ assigns measure one to some singleton $\left\{ x\right\}$, we say that $B$ is a $d$-dimensional Brownian motion starting at $x$.
5.8 Definition. A $d$-dimensional Brownian family is an adapted, $d$-dimensional process $B=\left\{ B_{t},\mathscr{F}_{t}:0\le t<\infty\right\}$ on a measurable space $\left(\Omega,\mathscr{F}\right)$, and a family of probability measures $\left\{ P^{x}\right\} _{x\in\mathbb{R}^{d}}$, such that
• (i) for each $F\in\mathscr{F}$ the mapping $x\mapsto P^{x}\left(F\right)$ is universally measurable;
• (ii) for each $x\in\mathbb{R}^{d}, P^{x}\left\{ B_{0}=x\right\} =1$;
• (iii) under each $P^{x}$, the process $B$ is a $d$-dimensional Brownian motion starting at $x$.
5.11 Definition. Let $d$ be a positive integer. A $d$-dimensional Markov family is an adapted process $X=\left\{ X_{t},\mathscr{F}_{t}:0\le t<\infty\right\}$ on some $\left(\Omega,\mathscr{F}\right)$, together with a family of probability measures $\left\{ P^{x}\right\} _{x\in\mathbb{R}^{d}}$ on $\left(\Omega,\mathscr{F}\right)$, such that
• (a) for each $F\in\mathscr{F}$ the mapping $x\mapsto P^{x}\left(F\right)$ is universally measurable;
• (b) for each $x\in\mathbb{R}^{d}$, $P^{x}\left\{ X_{0}=x\right\} =1$;
• (c) for each $x\in\mathbb{R}^{d}$, $s,t\geq0$ and $\Gamma\in\mathscr{B}\left(\mathbb{R}^{d}\right)$, $$P^{x}\left[X_{t+s}\in\Gamma|\mathscr{F}_{s}\right]=P^{x}\left[X_{t+s}\in\Gamma|X_{s}\right],\quad P^{x}\text{ a.s.}$$
• (d) for each $x\in\mathbb{R}^{d}$, $s,t\geq0$ and $\Gamma\in\mathscr{B}\left(\mathbb{R}^{d}\right)$,$$P^{x}\left[X_{t+s}\in\Gamma|X_{s}=y\right]=P^{y}\left[X_{t}\in\Gamma\right],\quad P^{x}X_{s}^{-1}\text{a.e. }y\in\mathbb{R}^{d}$$
Now I want to prove: 5.12 Theorem. A $d$-dimensional Brownian family is a Markov family. So we just need to show the condition (c) and (d). Assume a $d$-dimensional Brownian family $X=\left\{ X_{t},\mathscr{F}_{t}:0\le t<\infty\right\}$ on a measurable space $\left(\Omega,\mathscr{F}\right)$, and a family of probability measures $\left\{ P^{x}\right\} _{x\in\mathbb{R}^{d}}$.
Now (c) has already been proved. To prove (d), under each $P^{x}$, the process $X$ is a $d$-dimensional Brownian motion starting at $x$, so
$$P^{x}\left[X_{t+s}\in\Gamma|X_{s}=y\right]=P^{x}\left[\left(X_{s+t}-X_{s}\right)+y\in\Gamma\right],\quad\text{for }P^{x}X_{s}^{-1}\text{a.e. }y\in\mathbb{R}^{d}$$ Then
$$P^{x}\left[\left(X_{s+t}-X_{s}\right)+y\in\Gamma\right]=P^{x}\left[X_{t}+y\in\Gamma\right]$$
But how can I show that $P^{x}\left[X_{t}+y\in\Gamma\right]=P^{y}\left[X_{t}\in\Gamma\right]$?
Thanks a lot in advance!
Updates:
Now I think the definition of the Markov family is wrong. The condition (d) should be
• (d) for each $s,t\geq0$ and $\Gamma\in\mathscr{B}\left(\mathbb{R}^{d}\right)$,$$P^{0}\left[X_{t+s}\in\Gamma|X_{s}=y\right]=P^{y}\left[X_{t}\in\Gamma\right],\quad P^{0}X_{s}^{-1}\text{a.e. }y\in\mathbb{R}^{d}$$ rather than $P^{x}$. Because if we use $P^{x}$, we will have $$P^{x}\left[X_{t}+y\in\Gamma\right]=P^{y}\left[X_{t}\in\Gamma\right]$$ for all $x,y$, which I think is impossible.
But I can not find the definition of the Markov family in other contexts. Can anyone help me?
There are some mistakes in your post. In fact, $$P^{x}\left[\left(X_{s+t}-X_{s}\right)+y\in\Gamma\right]=P^{x}\left[\left(X_{t}-X_{0}\right)+y\in\Gamma\right]$$ Under $P_{x}$, $X_{t}-X_{0}$ is normally distributed with mean zero and covariance matrix equal to $tI_{d}$, hence $\left(X_{t}-X_{0}\right)+y$ is normally distributed with mean $y$ and covariance matrix equal to $tI_{d}$. Under $P_{y}$, $X_{t}-X_{0}$ is also normally distributed with mean zero and covariance matrix equal to $tI_{d}$, and $P^{y}\left\{ X_{0}=y\right\} =1$. Thus $X_{t}=X_{t}-X_{0}+X_{0}$ is normally distributed with mean $y$ and covariance matrix equal to $tI_{d}$. Hence $$P^{x}\left[\left(X_{t}-X_{0}\right)+y\in\Gamma\right]=P^{y}\left[X_{t}\in\Gamma\right]$$ hence $$P^{x}\left[X_{t+s}\in\Gamma|X_{s}=y\right]=P^{y}\left[X_{t}\in\Gamma\right],\quad\text{for }P^{x}X_{s}^{-1}\text{a.e. }y\in\mathbb{R}^{d}$$