CAPACITY ESTIMATES, BOUNDARY CROSSINGS AND THE ORNSTEIN{UHLENBECK PROCESS IN WIENER SPACE

Let $T_1$ denote the first passage time to 1 of a standard Brownian motion. It is well known that as $\lambda$ goes to infinity, $P\{ T_1 > \lambda \}$ goes to zero at rate $c \lambda^{-1/2}$, where $c$ equals $(2/ \pi)^{1/2}$. The goal of this note is to establish a quantitative, infinite dimensional version of this result. Namely, we will prove the existence of positive and finite constants $K_1$ and $K_2$, such that for all $\lambda>e^e$, $$K_1 \lambda^{-1/2} \leq \text{Cap} \{ T_1 > \lambda\} \leq K_2 \lambda^{-1/2} \log^3(\lambda) \cdot \log\log(\lambda),$$ where `$\log$' denotes the natural logarithm, and $\text{Cap}$ is the Fukushima-Malliavin capacity on the space of continuous functions.


Introduction
The goal of this note is to present a capacitarian extension of the classical fact that lim λ→∞ λ 1/2 P{T 1 > λ} = (2/π) 1/2 , where T 1 is the first passage time to 1 of a standard linear Brownian motion B = {B(t); t ≥ 0}.
Let Ω = C([0, ∞)) denote the collection of all continuous real functions on [0, ∞). As usual, Ω is made into a Banach space, once it is endowed with the supremum norm. Let F denote the collection of all of its Borel sets and let W denote Wiener's measure on (Ω, F ). The probability triple (Ω, F , W ) is the classical Wiener space, and let O = O s ; s ≥ 0 denote an Ornstein-Uhlenbeck process on (Ω, F , W ), which is an Ω-valued diffusion with stationary measure equal to W and whose increments are independent one-dimensional Ornstein-Uhlenbeck processes.
Williams [5] has observed that O can be described path-by-path, using a two-parameter Brownian sheet W = {W (s, t); s, t ≥ 0}. Namely, we can define O s for each s as the random function O s (t) = e −s/2 W (e s , t), t ≥ 0.
By Fukushima-Malliavin capacity, we mean the following: for all Borel sets A ⊂ Ω, This is also called the 1-capacity of A, as it is related to the 1-potential measure of O. The following is the main result of this paper. Theorem 1.1 There exist positive and finite constants K 1 and K 2 such that for all λ > e e ,
There is a relation to the recent results of Csáki, Khoshnevisan and Shi [1]. Namely, by Lemma 2.2 below, and stated in terms of the observation of D. Williams, Theorem 1.1 asserts the existence of finite and positive constants K 3 and K 4 , such that for all λ > e e , Above and hereafter, we designate uninteresting constants by K, K 5 , K 6 , . . .. These may change from line to line as well as within the lines.

Background Estimates
In this section, we present two basic estimates. For this first estimate, let U = {U (x); x ∈ R} denote an Ornstein-Uhlenbeck process that is indexed by R and is speeded up so that U is a centered Gaussian process with covariance E{U(x)U(y)} = e −|x−y| , x,y∈ R.
(2.1) Lemma 2.1 There exist two finite constants x 0 ∈ (0, 1) and t 0 > 0, such that for all x ∈ (0, x 0 ) and all t > t 0 , whose symmetrizing measure is the standard Gaussian. Thus, a routine application of the spectral theorem shows that the probability in the statement of the lemma has an eigenfunction expansion in terms of the (countable) eigenvalues of the (compact operator) A. Ref. [4] contains all of the delicate information that we will need about these eigenvalues to which the reader is referred for further details. Let . . denote the ordered eigenvalues and the orthonormalized (in L 2 (e −t 2 /2 dt)) eigenfunctions of A on (−∞, x) with zero boundary conditions. Then, We will need the following three facts about these eigenvalues: The result follows from facts (iii) and (ii).
For all r ≥ 0 and for all Borel sets A ⊂ Ω, we define the incomplete r-capacity Cap r (A) as Our second background estimate relates capacities to incomplete capacities and is an exercise in Laplace transforms. We point out that this result has already been used in the Introduction to establish Eq. (1.2).

Lemma 2.2
There exists a finite constant K > 1, such that for all Borel sets A ⊂ Ω, Proof Clearly, This implies the lower bound. For the upper bound, note that By stationarity, Cap(A) ≤ Cap 1 (A) ∞ j=0 (j + 1)e −j , and the lemma follows.

The Proof of Theorem 1.1
Throughout this proof, B = {B(t); t ≥ 0} denotes a standard linear Brownian motion and ε stands for a small positive number. We will also need three variables all of which are functions of ε as follows: where c 0 ∈ (0, ∞) is chosen to satisfy By the law of the iterated logarithm, such a c 0 must exist and can be chosen independently of the values of δ and ε. Consider the following random time that is finite (a.s., but this is taken care of in the usual way by adding in appropriate null sets): where W = {W (s, t); s, t ≥ 0} is a two-parameter Brownian sheet. Let F 1 denote the (complete, right continuous) filtration of the infinite-dimensional process {W (s, •); s ≥ 0}. It is easy to see that σ is a stopping time with respect to the one-parameter filtration F 1 . Next, we define two events E and F: Since {W (s, •); s ≥ 0} is a Lévy process on Ω, the following lemma can be easily verified: (ii) by the triangle inequality, E ∩ {σ ≤ a} ⊂ F.
The following lemma partly shows our interest in the event F. The event E is used in our derivation that is to come.
Proof By Lemma 3.1, on {σ ≤ a}, The last line follows from (3.3). Using Lemma 3.1 (ii), we can deduce On the other hand, inf The lemma follows.
To estimate P{F}, we observe that when ε is small, so that by scaling, Define the process U by U (x) = B(e −2x )/e −x , x ∈ R. It follows from direct covariance computations that U is the same (in law) as the Ornstein-Uhlenbeck process in (2.1). Moreover, Combining this with (3.4) and Lemma 2.1, we readily obtain the following: Lemma 3.2 and the stationarity of the Ornstein-Uhlenbeck process, imply the following result that is interesting in its own right.

Proposition 3.3
For all positive and finite K 5 , there exists a finite K 6 > 1, such that whenever I is an interval in [1, K 5 +1] whose length is bounded above by c 2 0 log 2 (1/ε) log log(1/ε) −1 , Proof of Theorem 1.1 Since [1, e] can be covered by 2c 2 0 log 2 (1/ε) log log(1/ε) many intervals I of the above type, we deduce the following estimate: for all ε > 0 small, It is possible to refine the rate given by Proposition 3.3, if the intervals are kept to small sizes. We conclude this article with a precise statement of this claim and its proof.

Proposition 3.4
For all positive and finite C 1 , there exists a finite C 2 > 1, such that whenever I is an interval in [1, 1 + C 1 ] whose length is at most C 1 ε 2 , By (1.1), this is sharp, up to a constant.
Proof Without loss of generality, where 1{· · ·} denotes the indicator function of the events in the parentheses. Since the elements of I are greater than 1, Eq. (1.1) implies that for all small ε > 0, where |I| = C 1 ε 2 denotes the length of I. We now compute a conditional version of this calculation. Recalling the 1-parameter filtration F 1 , we define the martingale M as a continuous modification of the following Observe that for all r ≥ 0, almost surely. Moreover, continuity considerations imply that the above holds a.s., simultaneously for all r ∈ [p, p + C 1 ε 2 /2]. By scaling, this leads to: Consider the F 1 stopping time T = inf{s ≥ p : sup 0≤t≤1 W (s, t) ≤ ε/2}, where inf ? = +∞.
Applying r ≡ T and taking expectations in the above to see that Since M is a bounded martingale, by the optional stopping theorem and by Eq. (3.6), E M T 1{T < ∞} = E{M 0 } = E{J} ≤ K 9 ε 3 . The proposition follows upon relabeling the parameters C 1 and C 2 .