Is the Hölder Norm of a Wiener Field a sub-Gaussian/sub-exponential variable?

Question

Let $W$ be a Wiener field over $[0,1]^d$ (see definition here : https://encyclopediaofmath.org/wiki/Wiener_field). I am interested in concentration bounds for the $\alpha$- Hölder norm (for $\alpha< 1/2$), define by, $$C_{\alpha}=\underset{(s_1,...,s_d),(t_1,...,t_d)\in[0,1]^d}{\sup}\quad\frac{|W_{s_1,...s_d}-W_{t_1,...t_d}|}{\|(s_1,...,s_d)-(t_1,...,t_d)\|^{\alpha}}$$ I know that $C_{\alpha}$ admits all its moments (see theorem 10.1, René L. Schilling and Lothar Partzsch., Brownian Motion). This gives me concentration bounds through Markov inequality. But I wonder if $C_{\alpha}$ is a sub-Gaussian variable or a sub-exponential variable, which would give me a sharper bound.