ON THE OCCUPATION TIME OF BROWNIAN EXCURSION

Recently, Kalvin M. Jansons derived in an elegant way the Laplace transform of the time spent by an excursion above a given level $a>0$. This result can also be derived from previous work of the author on the occupation time of the excursion in the interval $(a,a+b]$, by sending $b \to \infty$. Several alternative derivations areincluded.


Introduction
In [5], the author derives in an elegant way the Laplace transform of the time spent by an excursion above a given level a > 0. This result can also be derived from the occupation time of the excursion in the interval (a, a + b], by sending b → ∞ (cf. [2] or [4]).
Denote by W + 0 , Brownian excursion with time parameter t ∈ [0, 1], see [4], I.2 for a precise definition. According to p. 117 and p. 120 of [4], or Theorem 5.1 of [2], the Laplace transform of the occupation time T (a, a + b) = 1 0 1 (a,a+b] (W + 0 (t)) dt, is given by: where the path S is defined by In order to write the first term on the right side of (1), which term is equal to the distribution function of the supremum of Brownian excursion, 1 as a complex integral we introduce the path: with π/2 < φ < π, ξ > 0 and the orientation counterclockwise. We choose the angle φ in such a way that all sigularities of the integrand in (1) remain on the left of the path Γ.
since the integrand has only simple poles at α k = −k 2 π 2 /2a 2 , k ≥ 1. Combining (1) and (2) and deforming the path S into the path Γ (again using Cauchy's theorem), yields By taking the limit for b → ∞, (ψ(., ., b √ 2) → 1, uniformly on compacta of Γ) we obtain for the Laplace transform of the occupation time T (a) = T (a, ∞), Alternatively, one could take the limit for a ↓ 0 in (3), resulting in the transform: Ee −β(1−T (b)) . For the occcupation time T t (a) of the excursion straddling t, we have with T (a) and L t independent, and where L t denotes the length of the excursion. It is readily verified from the density of L t , see [1], (4.4), that for integrable ϕ, which is the more familiar form of this distribution function.
Hence, using (5) and (6), the Laplace transform (4) yields the double Laplace transform: This result can also be derived starting from reflected Brownian motion |W | (cf. [3], p. 92, Remark (3.20)). Perhaps the most elegant formulation of the Laplace transform of the occupation time is that for β strictly positive Equation (8) can be derived as follows. On the path Γ we have: dα.
Now for a > 0 the integral over the path Γ may be replaced by integration over the line (c − i∞, c + i∞), where c > 0 is arbitrary. Hence after the substitution α = xz, with x positive and replacement of the path (c/x − i∞, c/x + i∞) by the path (c − i∞, c + i∞), we obtain dz.
Taking Laplace transforms on both sides gives (8).