 Continuous Lévy process

A real Lévy process with continuous paths is Brownian motion with deterministic linear drift: $X_t=bt+\sigma W_t$. Continuity forces the jump measure in the Lévy–Khintchine formula to vanish, leaving characteristic function $\exp (t(ibu-{\sigma }^2{u}^2/2))$. If its law is also invariant under $X_t\mapsto {\lambda }^{-1}{X}_{{\lambda }^{2}t}$ for every $\lambda >0$, then $b=0$. The zero-variance case is included.