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.