The Kolmogorov-Chentsov continuity theorem is a general way to estimate the Hölder continuity of a process $X$ (up to taking a different version $\tilde{X}$ of $X$ ). However, I would like to know if this criterium is optimal. That is, given a process $X:[0,K] \longrightarrow \mathbb{R}$ define:
$$ \alpha := \sup\Big\{ \frac{\log \mathbb{E}[|X_t-X_s|^p]}{p\log |t-s|}-\frac{1}{p} \; \Big| \; t,s \in [0,K], p> 1, \mathbb{E}[|X_t-X_s|^p]<\infty \Big\} $$
Then, is it true that there exists no version $\tilde{X}$ of $X$ s.t $\tilde{X}$ is $\alpha$-Hölder? Or maybe, just for $\beta >\alpha$?
For instance, this statement is true for the Brownian Motion, however, at least in the proof I know for such fact uses the exact scaling $B_{at} \sim \sqrt{a}B_t$, however, the moment condition is more general. Is there an additional assumption you need to make the statement true?
I appreciate any references or ideas.
EDIT: I suppose the constant I am ignoring is actually going to make a difference. So let me rephrase it.
For $p > 1$, define $\alpha(p)$ via $$ \log \mathbb{E}|X_t - X_s|^p = \alpha(p)\log|t-s| + O(1). $$ if such expansion is possible. Now, define $\alpha_c = \sup\{ \frac{\alpha(p)-1}{p}: p \text{ s.t }\mathbb{E}[|X_t-X_s|^p] <\infty \}$. And then, the question is wether $X$ has a $C^\beta$ version for $\beta>\alpha_c$.
Consider a Poisson process with intensity $1$. Since $\mathbb{E}(|X_t-X_s|^2)=|t-s|$, it follows that $\alpha \geq 0$ and $\alpha_c \geq 0$. However, Poisson process does not have a continuous modification.
If you want to see sharpness of Kolmogorov-Chentsov theorem for continuous processes, then you can consider a deterministic function $f$ which is $\beta$-Hölder for $\beta<\alpha$ but not $\alpha$-Hölder-continuous, e.g. $\alpha=1$ and $$f(t) := t \log(|t|), \qquad t>0.$$ Then consider the deterministic process $X_t := f(t)$. You can easily check that $\alpha_c=1$ but $(X_t)_{t \geq 0}$ has no modification which is Lipschitz continuous.