Let $X$ be a $\mathbb R^d$ valued continuous stochastic process. I am interested in bounding $$P(\|X\|\_\gamma>R).$$


The standard technique to do so, is to apply Markov inequality and then Garsia-Rodemich-Rumsey inequality. Recall that GRR inequality says that



> 
> For any $\alpha>1$, $\delta>1/\alpha$ there is some constant $C:=C(\alpha, \delta)$ so that for all continuous $f\in C([0,1],\mathbb R^d)$ and $s,t\in [0,1]$ we have $$|f(t)-f(s)|^\alpha\leq C |t-s|^{\delta\alpha-1}\int\_s^t \int\_s^t \frac{|f(u)-f(v)|^\alpha}{|u-v|^{\delta\alpha + 1}}du\,dv.$$
> 
> 
> 


If we have the Kolmogorov type bound $E[|X(t)-X(s)|^\alpha]\leq K|t-s|^{1+\beta}$ and $\gamma<\beta/\alpha$ we may follow the answer [here](https://mathoverflow.net/questions/278977/kolmogorov-continuity-theorem-and-holder-norm/279085#279085), and apply Markov to get that


\begin{align\*}
P(\|X\|\_\gamma>R)&=P(\|X\|\_\gamma^\alpha>R^\alpha)\\
&\leq \frac{E[\|X\|\_\gamma^\alpha]}{R^\alpha}\\
&\le \frac{1}{R^\alpha}C(\delta,\alpha) \int\_0^T\int\_0^T \frac{\mathrm{E} \left[|X(u)-X(v)|^\alpha\right]}{|u-v|^{\delta\alpha + 1}}du\,dv \\
 &\le \frac{1}{R^\alpha}C(\delta,\alpha) K\int\_0^T\int\_0^T |u-v|^{\beta-\delta\alpha}du\,dv \\
&= \frac{1}{R^\alpha} C(\delta,\alpha,\beta,T)K.
\end{align\*}


However, Markov in general is a pretty crude bound. Furthermore, $X$ may not have too many moments. I am wondering if we only have tail bounds of the form


$$P\left(\frac{|X(t)-X(s)|}{|t-s|^p}>R\right)\leq \Phi(R)$$


for some decreasing function $\Phi$, what can we say? Assume $\Phi$ is continuous if necessary.


That is, can we express


$$P\left(\int\_0^T\int\_0^T \frac{|X(u)-X(v)|^\alpha}{|u-v|^{\delta\alpha + 1}}du\,dv>R^\alpha\right)$$


in terms of $\Phi$?



Garsia-Rodemich-Rumsey without Markov
Apologies for the earlier error. Upon examining what went wrong, I think I have discovered that no such bound can hold - there is in general no relationship between $\Phi$ and the Holder norm of $X$.


The idea is to define $X$ to be a small bump placed uniformly at random. Changing the parameters of this bump gives us arbitrarily nice functions $\Phi$ but arbitrarily bad Holder norms simultaneously.


More precisely, for any fixed positive integer $n \geq 22$, let $X^n$ be defined as follows - uniformly at random pick a point $p$ in, say $[\frac{1}{4}, \frac{3}{4}]$, and define


$$
X^n :=
\begin{cases}
0 & \text{on } [0, p], \\
n(t - p)^\gamma & \text{on } (p, p + \frac{1}{n^{2/\gamma}}], \\
\frac{1}{n} & \text{on } (p + \frac{1}{n^{2/\gamma}}, 1]. \\
\end{cases}
$$


Then it can be computed that for every fixed $ R$,


$$\sup\_{t \neq s} \mathbb P \left(\frac{|X^n(t)-X^n (s)|}{|t-s|^\gamma}>R\right) \to 0$$


as $n \to \infty$, but $\| X^n \|\_{\gamma} = n$ almost surely, which of course tends to $\infty$.


