We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too.
Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$ and I seek another Brownian motion, $W^*$ such that $W_t=W^*_{t+t_0}-W^*_{t_0}$ does such Brownian motion exists ?