On the local time density of the reflecting Brownian bridge

Expressions for the multi-dimensional densities of Brownian bridge local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for strata of random mappings.

Apart from the random mapping (see Section 2.3), applications of the rBB local time can be found in the analysis of Shellsort (see Louchard [17,2]).We need the distribution of the number I(2n) of inversions in a 2-ordered permutation of the 2n values {1...2n} (i.e., permutation consisting of two interleaved sorted permutations).
Position of the odd part of the permutation contains value k if the path U n corres- ponding to the permutation satisfies Un(i k-(this is Knuth's correspondence be- tween 2-ordering and path in a lattice, see Knuth [15,p. 87]).Now Vn([nt]nt)/x/rBB(t) and I(2n) ?=,l_[i-Un(i 1 |_ The local time of the rBB at /x/2n corresponds to the number of positions con- taining 2i + The rBB local time corresponds also to the number of jumps at some level of the empirical distribution, in the context of the classical Kolmogorov-Smirnov distribu- tion function. Denote by fx(Y) the density of r + (x).Then in [7] we find the representation fx(Y) l/-x-exp Y S V/ cosh(x) da (1) where S: (1-ioc, a + ice), a > 0, is a straight line parallel to the imaginary axis, which is the Brownian bridge analog to the density presented in [5, 13] for the Brown- ian excursion.We will generalize this formula to several dimensions and offer two approaches: The first one is a direct computation by means of Kac's formula for Brownian functionals and the second one is based on the fact that the process consist- ing of the-suitably normalized-strata of a random mapping converges weakly to Brownian bridge local time.
The paper is organized as follows.In Section 2, we summarize basic notations and methods.Some preliminary formulas based on Kac's formula and their inversion are given in Section 3. Section 4 is devoted to the general multi-dimensional density.
The Brownian excursion analog of this problem has been treated in [12].Thus we keep our presentation rather brief and refer to [12] for details.
We would like to mention that MAPLE was of great help in computing some com- plicated expressions (with some guidance, of course).

Basic Notations and Known Results
2.1 Kac's formula for Brownian functionals Denote by q(t) any of the processes defined in the previous section.
the notation E a[B(rl)]: Pr[B r/(O) a] Then we will use where B(r/) is an event belonging to the Borel field generated by 7(t).Furthermore, denote by c(f(x)):-fe-czf(x)dx, the Laplace transform of f(x).Then the classical density (for q(t)= x(t)) p(t x,y)dy" Ex[x(t dy] 1 exp/ (x- dt dy implies Za(p(t,x,y)) where the Laplace transform is taken with respect to t.
Let h >_ 0 be a piecewise continuous function and let be the differential operator (u)(a)" --ul ,,(a) h(a)u(a).
Kac's formula states that, for c > 0 and f E C(R1), u(a) E a / e-Ctexp h[x(s)]ds f(x(t))dt 0 0 is the bounded solution of (a )u f. (4) The solution of (4) is given by u(a)-f G(a,b)f(b)db where the Green function G is given by G(a, b) G(b, a) 2W l gl (a)g2(b), a <_ b, where 0 < gl E T, 0 < g2 G are independent solutions of g ag and W is their con- stant positive Wronskian: W g'lg2-glg'2 (see It6 and McKean [14, par. 2.6] and Louchard [18]).

Random mappings and local time
As usual, a random mapping on the set {1,...,n) is defined to be an element of the set F n of all mappings o:{1,...,n}--,{1,...,n) equipped with the uniform distribu- tion.It can be represented by its functional graph G,, i.e., the graph consisting of the nodes 1,2,...,n and of the edges (i,(i)), i= 1,...,n.It is easy to see that each component of such a graph consists of exactly one cycle of length _> 1, each point of which is the root of a labeled tree (a so-called Cayley tree).Thus for each point x E G, there exists a unique path connecting x with the next cyclic point.The length of this path is called the distance of x to the cycle.The set of all points at a fixed dis- tance r from the cycle is often called the rth stratum of o.
Let Ln(t denote the number of nodes in the tth stratum of a random mapping o F n.For noninteger t, define L() (tJ + 1 )L(EJ) + (-LJ + 1), >_ o.
There is a lot of literature on random mappings and interested readers should consult [16].In the sequel, we will need the following results from [7].

Preliminary Formulas
In this section, we define some auxiliary functions built on Kac's formula.Let (Pi) be a strictly monotonically increasing sequence of non-negative real numbers.Set dt and where m is related to the Brownian bridge of duration t.
In order to get our density representations we have to invert the formulas.This is done by the following.

Multi-dimensional Densities
We offer two different proofs of our results: The first one is based on some properties of G*.The other one is based on Cauchy's formula applied to (8) and singularity analysis.
4.1 The Proofs 4.1.1Using some properties of G* Proof of Lemma 3.3 Part (A): Actually, we will use the same notation as in the proof of Lemma 3 in [12] and use auxiliary functions Da(d), n4(d), for which we will prove the following relations: The coefficient of/1" "Zd 3 in D4(d 2 (d -1)/2Chl YI = 2Shl, l-1.
The second part gives ad-1 4.1.2Using the random mapping approach Now we will use the results of Section 2.2 in order to deduce Theorem 4.1.The proof runs in the following way: First we apply Lemma 2.1 of [6] in order to get an asymp- totic expansion of g(z, Ul,... Ud).Then we will apply Cauchy's formula and singulari- ty analysis in the sense of Flajolet and Odlyzko [9].We will omit details like error estimates, since this works in a very similar way as in [12].There is also another way to get a more rigorous proof via the random mapping approach: When we consider random mapping built of planted plane trees instead of Cayley trees.Since this can be viewed as a special case of constrained random mappings (see [3,4,11]), it is easy to see that Theorem 2.1 still holds (with a different scaling parameter of course: instead of 2).Thus the explicit formulas [12, Equations ( 31) and ( 32)] can be used instead of the asymptotic ones below and the error estimates are much easier.But on the other hand, dealing with those explicit expressions is much raore involved and does not provide any deeper insight.
We have the following lemma.