## Abstract

The joint law of the integral and the maximum of a Brownian excursion *Y* with a single mark of an independent Poisson point process of rate is determined. We use the identity in law of *Y* and two independent Brownian motions with drift −*u* (*u*>0) joined back-to-back with starting point having an exp(2*u*) distribution and stopped on hitting 0. This characterization of a Brownian excursion is expected to lead to a much easier starting point for calculating many other explicit joint laws associated with Brownian excursions.

## 1. Introduction

We begin by considering *X*=(*X*_{t}, −∞<*t*<∞) a standard Brownian motion with drift −*u* (*u*>0), with values in reflected at 0. The excursions of *X* from 0 are related to those of Brownian motion, and are in fact Brownian excursions conditioned on not being marked by the points of an independent Poisson point process of rate Rogers & Williams (1987).

Let *X*^{0} denote the excursion of *X* that includes *t*=0. *X*^{0} can be reconstructed as follows. has an exp(2*u*) distribution and *X*^{0} restricted to *t*>0 is a Brownian motion with drift −*u* (*u*>0) started at and stopped when it hits 0. Also, inherited from *X*, *X*^{0} is preserved under time reversal, namely, given ,By establishing a connection between *X*^{0} and a Brownian excursion, we are able to calculate the joint law of the integral and the maximum of a Brownian excursion. The method used is applicable to many similar problems involving Brownian excursions and has applications to Gaussian ring polymers (see Jansons (1997)).

## 2. Brownian excursions with a single mark

We shall use the semi-martingale normalization of excursion rates, as used, for example, in Rogers & Williams (1987). In this section we derive repackaged versions of some results from Jansons (1997), but in a form that is much more convenient for applications, e.g. in polymer physics.

*Suppose that X*=(*X*_{t},−∞<*t*<∞) *with values in* [0,∞) *is a standard Brownian motion with drift* −*u* (*u*>0) *reflected at* 0. *The excursion X*^{0} *of X that contains t*=0 *is identical in law to an excursion of a standard Brownian motion conditioned on having a single mark of an independent Poisson point process of rate* .

The excursions of *X* are identical in law to the excursions of standard Brownian motion conditioned on not being marked by an independent Poisson point process of rate .

Let *n*_{u}{.} denote the excursion rate for *X*, and *n*_{+}{.} denote the excursion rate of positive excursions of standard Brownian motion.where *T* is the duration of the excursion. Let **P**_{0}[.] denote the probability measure of *X*^{0}. Conditioned on the duration *T* of the excursion,As the law of *X* is invariant under time translation, for given duration *T*, the excursion containing *t*=0 has starting point uniformly distributed in (−*T*, 0); thusFrom above, we findwhich is the law of a Brownian excursion with a single mark, orwhere the constant follows from , determined previously (see Jansons (1997)). ▪

A more natural version of lemma 2.1 for future work is given below in Theorem 2.2.

*A positive excursion of a standard Brownian motion with a single mark of an independent Poisson point process of rate* *can be decomposed at the mark into two independent standard Brownian motions with drift* −*u*(*u*>0) *stopped on hitting* 0 *joined back-to-back with the marked point having an* exp(2*u*) *distribution*.

This follows directly from lemma 2.1 and the observations about *X*^{0} in the introduction once we have identified the single mark on the Brownian excursion with the point *t*=0 of *X*^{0}. Both of these points are uniformly distributed with respect to time in the excursion and are otherwise independent of it. ▪

*Let A be an event that can be decomposed into A _{−} and A_{+}, which respectively depend on the t<*0

*and the t≥*0

*parts of X*

^{0}

*, then*

*where*

**[.]**

*Q**is the probability measure for a*-

*marked standard Brownian motion stopped on hitting*0

*and*

*is the time-reversal of A*

_{−}.

Since the probability density function of is 2*u*exp(−2*ux*), we knowAlso, as observed in the introduction,from which we findButandwhich appears in Rogers & Williams (1987). ▪

## 3. The joint law of the integral and the maximum of a Brownian excursion

As a sample application of the results of the previous section, we consider the joint law of the integral and the maximum of a Brownian excursion. This result has applications to Gaussian ring polymers as discussed later, which was the author's motivation, though the result is of probabilistic interest in its own right. A large number of similar results with ring-polymer applications can be obtained by the same method, some of which will appear elsewhere.

This demonstrates how excursion theory provides a very quick and easy way to determine the joint laws required in many applications.

*The joint law of the maximum and the integral of a positive excursion Y of a standard Brownian motion conditioned on having a single mark of an independent Poisson point process of rate* *is**where λ>*0*, I is the support of Y _{t}, and*

*The duration of Y has a* *distribution*.

Let *B*=(*B*_{t}, 0≤*t*) be a standard Brownian motion with values in [0,*a*] stopped on hitting 0 or *a*, and definewhere *ψ* satisfiesSolving for *ψ*, we find thatwhere in terms of Airy functions*M* is a bounded local martingale, hence a true martingale, with respect to *B*, thuswhere *H*_{0} is the time at which *B* hits 0.

Applying lemma 2.1, to identify *Y* with *X*^{0}, and lemma 2.3 we findwhere *I* is the support of .

The duration of *Y* is easily shown to have a distribution, and has been determined previously in Jansons (1997). ▪

## 4. Discussion

Theorem 2.2 provided an alternative, and preferable, starting point for some previous results involving Brownian excursions, including one by the author Jansons (1997). Theorem 3.1 extends that of Kennedy (1976), who found the law of the maximum of a Brownian excursion.

In principle we could obtain the corresponding result for a standard Brownian excursion (or a BES(3)BR process) by unmixing the distribution for the duration of *Y*, but it is unlikely that this could be done explicitly. However, the integral over [0,*a*] in theorem 3.1 can be done explicitly, though the result is a very messy expression in terms of hypergeometric functions.

As in previous work (Jansons (1997)), properties of Brownian excursions have direct applications to polymer rings via a theorem of Vervaat (1979). The expression in theorem 3.1 can be interpreted as the partition function for a Gaussian ring polymer in a linear potential near an impenetrable boundary. For more information on the connection between Brownian motion and polymer physics see Jansons & Rogers (1991).

## Footnotes

- Received November 26, 2004.
- Accepted August 1, 2005.

- © 2005 The Royal Society