Let us consider a generalized O-U process $X\_t \in L^2[0, 1]$ defined by the following spde: 


$dX\_t = \frac{1}{2}\partial\_x^2X\_t + dW\_t, $


$\partial\_x X\_t(0) = \partial\_x X\_t(1) = 0, $


$X\_0 = 0, $


where $W$ is a space-time white noise. Let $\{\varphi\_j\}$ be a CONS of $L^2[0, 1]$. I want to estimate the following amount for each $T>0$ and $\eta>0$: 
$$\lim\_{k\rightarrow \infty}\limsup\_{\epsilon \downarrow 0} P\left(\sup\_{t\in[0,T]}\sum\_{j=k}^\infty \langle \epsilon X\_{t\epsilon^{-2}}, \varphi\_j\rangle^2 > \eta\right). $$


Intuitively, if we have $\sum\_{j=1}^\infty E \left[\sup\_{t \in [0, T]}\langle X\_t, \varphi\_j\rangle^2\right] \sim T$, or simply $E \left[\sup\_{t\in[0,T]}||X\_t||^2\right] \sim T$
then the limit above shall be zero. 


It is enough to consider only O-U processes, not generalized ones. Given a collection of 1-dim O-U process $X\_t^j$ as: 


$dX\_t^j = -\lambda\_j X\_t^j dt + dB\_t^j, $


$X\_0^j = 0, $


where $B^j = W(\varphi\_j)$ are independent Brownian motions and $-\lambda\_j = -\frac{1}{2}\pi^2j^2$ are eigenvalues of Laplace operator. Then what I want turns to be 
$$\sum\_{j=1}^\infty E\left[\sup\_{t\in[0,T]}[X\_t^j]^2\right] = O(T). $$


The only thing I know about this is the estimate mentioned in [Holly-Stroock 1977], written as 
$$E \left[ \sup\_{t\in[0,T]}\langle X\_t^j\rangle^2 \right]<C(1+T)^3(1+\lambda\_j^2), $$
which is even not decay with $j$. 


Sorry for the long question and thank you very much. 



On the superior of generalized Ornstein-Uhlenbeck process
You won't have much luck with $\sum\_j\mathbb{E}[\sup\_T\langle X\_t,\varphi\_j\rangle^2]$, since this is not expected to converge. For simplicity say $\varphi\_j$ are the eigenvectors of the Laplace operator, so that $\langle X\_t,\varphi\_j\rangle$ is your $X^j$. Rescale $Y^j\_t = X^j(t/\lambda\_j)/\sqrt{\lambda\_j}$ to see that if $\mathbb{E}[\sup\_T|Y^j|^2]\approx c\_0T$, then $\mathbb{E}[\sup\_T|X^j|^2]\approx c\_0T$.


On the other hand you'll have better luck with $\mathbb{E}[\sup\_T\|X\_t\|^2]$ (I guess $\|\cdot\|$ here is the $L^2$ norm), or even better with $\mathbb{E}[\sup\_{t\leq T,x\in(0,1)}|X\_t(x)|]$ (it does not make any difference for your purposes). This expectation can be estimated using Corollary 4.15 of [this book](http://projecteuclid.org/euclid.lnms/1215467924) (for a computation very close to the one you need and that uses the same corollary, see for instance Lemma 2.1 of [this paper](http://www.numdam.org/item?id=AIHPB_1995__31_1_81_0)). Now it is just a matter of doing the required computations.


