In the book "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve I encountered Problem 6.1 which the authors refer to as "not be hard to show":
Let $\{B_t, \cal{F}_t;t\geq 0\}$ be a standard, one-dimensional Brownian motion. Give an example of a random time $S$ with $P[0\leq S<\infty]=1$, such that with $W_t:=B_{S+t}-B_S$, the process $W=\{W_t, \cal{F}^W_t;t\geq 0\}$ is not a Brownian motion.
Surely, if S being a stopping time $(W_t)_t$ would still be a Brownian motion. I tried out some distributions for $S$ but none of them gave a sufficient answer.
Most likely I am missing a crucial point here and appreciate any kind of help.
Let $S = \inf \{s\ge 0:B_{s+1}-B_s\ge 0\}$. Since $W_1\ge0$ almost surely, $W_t$ cannot be a Brownian motion.