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We explicitly calculate the Hausdorff dimension of the graph and range of anisotropic stable L\'{e}vy process $X$ plus deterministic drift function $f$.For that purpose we use a restricted version of the genuine Hausdorff dimensionwhich is called the parabolic Hausdorff dimension. It turns out that covers byparabolic cylinders are optimal for treating self-similar processes, sincetheir distinct non-linear scaling between time and space geometrically matchesthe self-similarity of the processes. We provide explicit formulas for theHausdorff dimension of the graph and the range of $X+f$. In sum the parabolicHausdorff dimension of the drift term $f$ alone contributes to the Hausdorffdimension of $X+f$. Further, we derive some formulas and bounds for theparabolic Hausdorff dimension.

Parabolic Fractal Geometry of Stable L\'evy Processes with Drift