Hausdorff and Fourier dimension of graph of continuous additive processes

Abstract

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as Xt=BV(t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity.It is also possible to estimate the Hausdorff dimension of the graph through the Lq spectrum of V. We will show that if V is generated by a self-similar measure on R1 with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its Lq spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is 1+12⋅log2log3. Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely.