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In the theory of extreme values of Gaussian processes, many results areexpressed in terms of the Pickands constant $\mathcal{H}_{\alpha}$. Thisconstant depends on the local self-similarity exponent $\alpha$ of the process,i.e. locally it is a fractional Brownian motion (fBm) of Hurst index$H=\alpha/2$. Despite its importance, only two values of the Pickands constantare known: ${\cal H}_1 =1$ and ${\cal H}_2=1/\sqrt{\pi}$. Here, we extend therecent perturbative approach to fBm to include drift terms. This allows us toinvestigate the Pickands constant $\mathcal{H}_{\alpha}$ around standardBrownian motion ($\alpha =1$) and to derive the new exact result$\mathcal{H}_{\alpha}=1 - (\alpha-1) \gamma_{\rm E} + \mathcal{O}\!\left(\alpha-1\right)^{2}$.

Pickands' constant at first order in an expansion around Brownian motion