Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 27J / b / Solution

The scaled process starts at zero and has continuous paths. Disjoint time intervals map to disjoint intervals under $t\mapsto c^{2}t$, preserving independence of increments, andThis holds for either sign of $c\ne 0$. Hence $\boxed{{\tilde{B}}_t=c^{-1}{B}_{c^{2}t}\text{ is standard Brownian motion}}$.