Hausdorff dimension of the contours of symmetric additive Lévy processes, Probab. Theory Related Fields

Abstract. Let X1,..., XN denote N independent, symmetric Lévy processes on Rd. The corresponding additive Lévy process is defined as the following N-parameter random field on Rd: (0.1) X(t): = X1(t1) + · · ·+XN (tN) (t ∈ RN+). Khoshnevisan and Xiao (2002) have found a necessary and sufficient condition for the zero-set X−1({0}) of X to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of X−1({0}) which hold with positive probability in the case that X−1({0}) can be non-void. Here we prove that the Hausdorff dimension of X−1({0}) is a constant almost surely on the event {X−1({0}) 6 = ∅}. Moreover, we derive a formula for the said constant. This portion of our work extends the well known formulas of Horowitz (1968) and Hawkes (1974) both of which hold for one-parameter Lévy processes. More generally, we prove that for every nonrandom Borel set F in (0,∞)N, the Hausdorff dimension of X−1({0}) ∩ F is a constant almost surely on the event {X−1({0}) ∩ F 6 = ∅}. This constant is computed explicitly in many cases. 1