Do we have Arzela-Ascoli theorem on the space of all continuous stochastic process?

Question

Let $\Omega$ be a polish space and $(\Omega,\mathcal{F},\mathbb{P})$ be a complete filtered probability space with the filtration generated by the standard Brownian motion. Let the space $U:=L^2\big(\Omega:C([0,T]:\mathbb{R})\big)$ with the norm $\|\cdot\|_U=\mathbb{E}(\sup_{s}|X_s|^2)$. Let $\{X_{k,s}\}\subset U$ such that $\sup_k\|X_{k,\cdot}\|_U \leq 1$ and $\mathbb{E}(|X_{k,s}-|X_{k,t}|^2)<|t-s|^\delta$ for some $\delta > 0$. Do we have the Arzela-Ascoli theorem such that there is some process $X^*_s$ such that $\|X_{k,\cdot}-X^*_\cdot\|_U\to0$ up to a subsequence?

Answer 1

$X_k$ need not have a subsequence that converges in $U$. One way to generate a counterexample is to consider constant processes (i.e. processes such that $t \mapsto X_{k,t}(\omega)$ is constant for each $\omega \in \Omega$).

Since $L^2(\Omega)$ is an infinite dimensional Banach space, by an application of Riesz' Lemma there exists a sequence $Y_k \in L^2(\Omega)$ such that $\|Y_k\|_{L^2(\Omega)} = 1$ and $\|Y_k - Y_j\|_{L^2(\Omega)} > 1/2$ for each $k \neq j$. Consider the constant processes $X_{k,t} = Y_k$.

These processes are easily seen to satisfy both of your conditions but since $$ \|X_k - X_j \|_U = \|Y_k - Y_j\|_{L^2(\Omega)} > 1/2 $$ for any $k \neq j$, they do not admit a convergent subsequence.