Let us say that I have a sequence of random $L^p$-valued functions $I^{\varepsilon} \in C([0,T], L^p(\Omega))$ such that
$$\forall_{\varepsilon > 0}\ [[ I^{\epsilon}(t) - I^{\varepsilon}(s) ]]_p \leq (t-s)^{\alpha} + \varepsilon^{\alpha}$$
where $[[ \cdot ]]_p$ is a $L_p$ norm and $\alpha > 0$. Is it possible to say that $(I^{\varepsilon} )_{\varepsilon>0}$ admits a convergent subsequence in this space?
edit: additionally assume that $[[ I^{\varepsilon} ]]_p \leq \sqrt{p} (t^{\alpha} + \varepsilon^{\alpha})$, which is known to correspond to having Gaussian tails, so it is tight $$[[ I^{\varepsilon} 1_{I^{\varepsilon} > y }]]_p \to_{y \to \infty} 0 $$
No. Take $\Omega = \mathbb R$. And define $$ I^\epsilon(t) = \phi(t + \epsilon^{-1}), $$ where $\phi$ is supported on $(0,1)$.
To get the desired convergence, you also need compactness in the image space.