Self-similar process

Remarks

The Wiener process $W$ is self-similar with $H = 1/2$. Check: $(W_{at})_{t \ge 0}$ and $(\sqrt{a}\, W_t)_{t \ge 0}$ are both centred Gaussian processes with covariance $a \min(s, t)$, so they have the same finite-dimensional distributions. Brownian motion with drift, $X_t = W_t + t$, is not self-similar for any $H$. Indeed $E[X_{at}] = at$ and $E[a^H X_t] = a^H t$ force $H = 1$, while $\operatorname{Var}(X_{at}) = at$ and $\operatorname{Var}(a^H X_t) = a^{2H} t$ force $H = 1/2$. One often writes $X_{at} \overset{d}{=} a^H X_t$, where equality in law is meant for the whole process. Taking $t = 0$ gives $X_0 \overset{d}{=} a^H X_0$ for all $a > 0$, which forces $X_0 = 0$ almost surely.