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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.

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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 (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). Now it is just a matter of doing the required computations.

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  • $\begingroup$ Although the discussion in the book of Adler is limited in 1 dimensional, it is very close to what I want. Thanks very much for your kindly help. $\endgroup$ – gregarki khayal Jul 4 '14 at 12:34
  • $\begingroup$ @LuTokisaka what do you mean by 1 dimensional? the book deals with random fields, in your case you apply the theorems to the random field X(t,x), indexed by (t,x). This is why looking at the sup in space and time works nicely $\endgroup$ – marcoromito Jul 4 '14 at 20:25

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