Consider a probability space $(\Omega, \mathcal{F}, \mathbf{P})$,a standard(one-dimensional) Brownian motion or Wiener process,denoted by $B=\{B_t:t\in [0,\infty)\}$,is a stochastic process on $(\Omega, \mathcal{F}, \mathbf{P})$ satisfying the following properties:
1. $\mathbf{P}(\{\omega:B_0(\omega) = 0\}) = 1.$
2. For every choice of nonnegative real numbers $0 \le t_1<t_2<\cdots <t_{n-1}<t_{n}<\infty$, the increments $B_{t_1}-B_{t_0}, B_{t_2}-B_{t_2},..., B_{t_n} - B_{t_{n-1}}$ are mutually independent random variables.($B$ has independent increments )
3. For each $0\le s<t<\infty$,the increment $B_t - B_s\sim\mathcal{N}(0,t-s).$($B$ has stationary increments)
4. $\mathbf{P}(\{\omega: t \mapsto B_t(\omega) \textrm{ is not continuous at each $t$}\}) = 0.$
$\textbf{Exercise.}$ Show that
$\textbf{(a)}$.For any $\delta\in\mathbb{R}^{+},\Gamma\in\mathscr{B}(\mathbb{R}),$ $$\mathbf{P}\left(\underset{u\in [s,t]}{\text{inf}}B_{u}\in \Gamma \bigg| B_{t}=\alpha,B_{s}=\beta \right)=\mathbf{P}\left(\underset{u\in [s+\delta,t+\delta]}{\text{inf}}B_{u+\delta}\in\Gamma \bigg| B_{t+\delta}=\alpha,B_{s+\delta}=\beta\right),$$ where $0\le s<t<\infty;\alpha,\beta\in \mathbb{R}.$
$\textbf{(b)}$.For any $\Delta\in\mathbb{R},\Gamma\in\mathscr{B}(\mathbb{R}),$ $$\mathbf{P}\left(\underset{u\in [s,t]}{\text{sup}}B_{u}\in \Gamma \bigg| B_{t}=\alpha,B_{s}=\beta \right)=\mathbf{P}\left(\underset{u\in [s,t]}{\text{sup}}B_{u}\in\Gamma+\Delta\bigg| B_{t}=\alpha+\Delta,B_{s}=\beta+\Delta\right),$$ where $0\le s<t<\infty;\alpha,\beta\in \mathbb{R}.$
Below are some of my own thoughts：
Fixed $0\le t<u<s<\infty$ and $\alpha ,\beta\in \mathbb{R}$, the conditional distribution of $B_{u}$ given $B_{s}=\alpha,B_{t}=\beta$ is normal: $$\left(B_u\bigg| B_s = \alpha, B_t = \beta\right)\sim \mathcal{N}\left(\alpha+(\beta-\alpha)\frac{u-s}{t-s},\frac{(t-u)(u-s)}{t-s}\right)$$
From above, ${\color{Blue} {\textsf{the time-homogeneous property of $B$}}}$ and ${\color{Blue} {\textsf{the spatially homogeneous property of $B$ }}}$ acquired without much effort.
$\textbf{(c)}.{\color{Blue} {\textsf{The time-homogeneous property of $B$}}}$ :
For any $\delta\in\mathbb{R}^{+},\Gamma\in\mathscr{B}(\mathbb{R}),$ $$\mathbf{P}\left(B_{u}\in \Gamma \bigg| B_{t}=\alpha,B_{s}=\beta\right)=\mathbf{P}\left(B_{u+\delta}\in \Gamma \bigg| B_{t+\delta}=\alpha,B_{s+\delta}=\beta\right),$$ where $0\le s<u<t<\infty;\alpha,\beta\in \mathbb{R}.$
$\textbf{(d)}.{\color{Blue} {\textsf{The spatially homogeneous property of $B$ }}}$:
For any $\Delta\in\mathbb{R},\Gamma\in\mathscr{B}(\mathbb{R}),$ $$\mathbf{P}\left(B_{u}\in \Gamma \bigg| B_{t}=\alpha,B_{s}=\beta\right)=\mathbf{P}\left(B_{u}\in \Gamma+\Delta \bigg| B_{t}=\alpha+\Delta,B_{s}=\beta+\Delta\right),$$ where $0\le s<u<t<\infty;\alpha,\beta\in \mathbb{R}.$
$\\$ I want to deduce $\textbf{(a),(b)}$ from $\textbf{(c),(d)}$ respectively, but how to proof that for each $u\in \mathcal{U}$ (an uncountable set in $[0,\infty)$) have the same conditional distributions,then those "infimum & supremum" also have identical conditional distribution? By the way,I speculate that a random process $X=\{X_t:t\in [0,\infty)\}$(not limited to Brownian motion) with stationary independent increments,then $\textbf{(c)}$ and $\textbf{(d)}$ are very likely to be true,(however,the validity of $\textbf{(a)} $ and $\textbf{(b)}$ still requires further confirmation).