# Brownian motion increments

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 ?

Start with a Brownian motion $Y_t$, where the processes $W$ and $Y$ are independent. Take $$W^*_t = \cases{ Y_t & for 0 \le t \le t_0\cr Y_{t_0} + W_{t-t_0} & for t > t_0\cr}$$