A scaling proof for Walsh’s Brownian motion extended arc-sine law

We present a new proof of the extended arc-sine law related to Walsh’s Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams [12], in the 1-dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.

1 Introduction a) Recently, some renewed interest has been shown (see e.g.[9]) in the study of the law of the vector 1 (Ws∈I i ) ds; i = 1, 2, . . ., n , where (W s ) denotes a Walsh Brownian motion, also called Brownian spider (see [10] for Walsh's lyrical description) living on I = n i=1 I i , the union of n half-lines of the plane, meeting at 0.
For the sake of simplicity, we assume p 1 = p 2 = . . .= p n = 1/n, i.e.: when returning to 0, Walsh's Brownian motion chooses, loosely speaking, its "new" ray in a uniform way.
In fact, excursion theory and/or the computation of the semi-group of Walsh's Brownian motion (see [1]) allow to define the process rigorously.
Since (d(0, W s ); s ≥ 0), for d the Euclidian distance, is a reflecting Brownian motion, we denote by (L t , t ≥ 0) the unique continuous increasing process such that: t , . . ., A (n) t denote the random vector of the times spent in the different rays.In Section 2 we will state and prove our main Theorem concerning the distribution of − → A t for a fixed time.Section 3 deals with the general case of stable variables, First, we recall some known results and then we state and prove our main Theorem.Finally, Section 4 is devoted to some remarks and comments.

b) Reminder on the arc-sine law:
A random variable A follows the arc-sine law if it admits the density: Some well known representations of an arc-sine variable are the following: where N, N ∼ N (0, 1) and are independent, U is uniform on [0, 2π], T and T stand for two iid stable (1/2) unilateral variables, and C is a standard Cauchy variable.With (B t , t ≥ 0) denoting a real Brownian motion, two well known examples of arc-sine distributed variables are: a result that is due to Paul Lévy (see e.g.[6,7,13]).
c) This point gives some motivation for Section 3. From (2), one could think that more general studies of the time spent positive by diffusions may bring 2 independent gamma variables (this because N 2 and N2 are distributed like two independent gamma variables of parameter 1/2), or 2 independent stable (µ) variables.It turns out that it is the second case which seems to occur more naturally.We devote Section 3 to this case.

Main result
Our aim is to prove the following: Theorem 2.1.The random vectors − → A T /T for: t > s}; (iii) T = τ l , the inverse local times, have the same distribution.In particular, it is specified by the iid stable (1/2) subordinators: which yields that: where T j are iid, stable (1/2) variables.
The law of the right-hand side of ( 3) is easily computed, and consequently so is its lefthand side.We refer the reader to [2] for explicit expressions of this law, which for n = 2 reduces to the classical arc-sine law.
Proof of Theorem 2.1.a) Clearly, (ii) plays a kind of "bridge" between (i) and (iii).b) We shall work with α s , s ≥ 0 , the inverse of A s , s ≥ 0 .We then follow the main steps of [13] (Section 3.4, p. 42), which themselves are inspired by Williams [12]; see also Watanabe (Proposition 1 in [11]) and Mc Kean [8].A (j) t denotes the time spent in I j , for any j = 1.Since , and for every u, t ≥ 0, L 2 and invoking the scaling property, we can write jointly for all j's: Dividing now both sides by α (+) 1 and remarking that: τ 1 = τ 1 , we deduce: With the help of the scaling Lemma below, we obtain: I 1 may be replaced by I m , for any m ∈ {2, . . ., n}.Adding the m quantities found in (7) and remarking that: we get: which proves (3).Note that from ( 6), the latter also equals: Equality in law ( 4) follows now easily.Indeed, we denote by ν the Itô measure of the Brownian spider, and we have: where ν j is the canonical image of n, the standard Itô measure of the space of the excursions of the standard Brownian motion, on the space of the excursions on I j .Hence, with λ j , j = 1, . . ., n denoting positive constants: The latter, using (8) yields: which finishes the proof.
It now remains to state the scaling Lemma which played a role in (7), and which we lift from [13] (Corollary 1, p. 40) in a "reduced" form.

Reminder and preliminaries on stable variables
In this Section, we consider S µ and S ′ µ two independent stable variables with exponent µ ∈ (0, 1), i.e. for every λ ≥ 0, the Laplace transform of S µ is given by: Concerning the law of S µ , there is no simple expression for its density (except for the case µ = 1/2; see e.g.Exercise 4.20 in [3]).However, we have that, for every s < 1 (see e.g.[15] or Exercise 4.19 in [3]): We consider now the random variable of the ratio of two µ-stable variables: Following e.g.Exercise 4.23 in [3], we have respectively the following formulas for the Stieltjes and the Mellin transforms of X: Moreover, the density of the random variable X µ is given by (see e.g.[14,5] or Exercise 4.23 in [3]): or equivalently: where, with C denoting a standard Cauchy variable and U a uniform variable in [ 0, 2π ), = sin(πµ − U) U .

The case of 2 stable variables
We turn now our study to the random variable: Theorem 3.1.The density function of the random variable A is given by: Proof of Theorem 3.1.Identity ( 19) is equivalent to: Hence, (15) yields: We consider now a test function f and invoking the density (17) we have (ν = 1 µ > 1): Changing the variables z = 1 1+y ν , we deduce: where: and (20) follows easily.
In Figure 1, we have plotted the density function g of A, for several values of µ.Remark 3.2.Similar discussions have been made in [4] in the framework of a skew Bessel process with dimension 2 − 2α and skewness parameter p. Formula (20) is a particular case of formula in [4] for the density of the time spent positive (called f p,α in [4]).

The case of many stable (1/2) variables
In this Subsection, we consider again n iid stable (1/2) variables, i.e.: T 1 , . . ., T n , and we will study the distribution of: The following Theorem answers to an open question (and even in a more general sense) stated at the end of [9].
Theorem 3.3.The density function of the random variable A 1 is given by: P A (1) Proof of Theorem 3.3.
We first remark that, with C denoting a standard Cauchy variable, using e.g.(2): Hence, with f standing again for a test function, and invoking the density of a standard 1 , for several values of n.

Conclusion and comments
We end up this article with some comments: usually, a scaling argument is "one-dimensional", as it involves a time-change.Exceptionally (or so it seems to the authors), here we could apply a scaling argument in a multivariate framework.We insist that the scaling Lemma plays a key role in our proof.The curious reader should also look at the totally different proof of this Theorem in [2], which mixes excursion theory and the Feynman-Kac method.

Figure 1 :
Figure 1: The density function g of A, for several values of µ.

Figure 2 :
Figure 2: The density function h of A and (22) follows easily.

Figure 2 1 ) 1 ,Corollary 3 . 4 .
Figure 2 presents the plot of the density function h of A (1) 1 , for several values of n.

4 .
It follows from Theorem 3.3 by simply remarking that C (law)