Note: I'm not an expert on stochastic processes. Please use small words and speak real slow.

I'm reading a paper [1], which uses a notation for stochastic processes that doesn't seem to be standard.

Now we generalize the squared covariance and define the square of conditional covariance, given two real-valued stochastic processes $U(·)$ and $V(·)$. We obtain an interesting result when $U$ and $V$ are independent Weiner processes. First, to center the random variable $X$ in the conditional covariance, we need the following definition. Let $X$ be a real-valued random variable and ${U (t):t ∈ ℝ}$ a real-valued stochastic process, independent of $X$. The $U$-centered version of $X$ is defined by

$$X_U = U (X) − \int^∞_{-∞} U (t) dFX(t) = U (X) − E[U (X)|U]$$

In my experience a stochastic process is a collection of random variables $\{X_t\}$... What is the intuition behind a stochastic process $U(·)$? Also, they have both $U(X)$ and $U(t)$ which I can't make heads or tails of.

Cheers