Continuous Lévy process
= Continuous Lévy process
{title2=$X_t=bt+\sigma W_t$}
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 $\exp(t(ibu-\sigma^2u^2/2))$. If its law is also invariant under $X_t\mapsto\lambda^{-1}X_{\lambda^2t}$ for every $\lambda>0$, then $b=0$. The zero-variance case is included.