  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 additive
processes with zero mean. Such processes can be represented as $X_t = B_{V(t)}$
where $B$ is Brownian motion and $V$ is a continuous increasing function. We
show that these dimensions depend on the local uniform H\"{o}lder 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
H\"{o}lder regularity.
  It is also possible to estimate the Hausdorff dimension of the graph through
the $L^q$ spectrum of $V$. We will show that if $V$ is generated by a
self-similar measure on ${\mathbb R}^{1}$ with convex open set condition, the
Hausdorff dimension of the graph can be precisely computed by its $L^q$
spectrum. An illustrating example of the Cantor Devil Staircase function, the
Hausdorff dimension of the graph is $1+\frac12\cdot\frac{\log 2}{\log 3}$.
Moreover, we will show that the graph of the Brownian staircase surprisingly
has Fourier dimension zero almost surely.
