The distribution of lo cal times of a Brownian
bridge
by
Jim Pitman
Technical Rep ort No. 539
Department of Statistics
University of California
367 Evans Hall 3860
Berkeley, CA 94720-3860
Nov. 3, 1998 1
The distribution of lo cal times of a Brownian bridge
Jim Pitman
Department of Statistics,
University of California,
367 Evans Hall 3860,
Berkeley, CA 94720-3860, USA
1 Intro duction
x
Let L ;t 0;x 2 R denote the jointly continuous pro cess of lo cal times of a standard
t
one-dimensional Brownian motion B ;t 0 started at B = 0, as determined bythe
t 0
o ccupation density formula [19 ]
Z Z
t 1
x
dx f B ds = f x L
s
t
0 1
for all non-negative Borel functions f . Boro din [7, p. 6] used the metho d of Feynman-
x
Kac to obtain the following description of the joint distribution of L and B for arbitrary
1
1
xed x 2 R: for y>0 and b 2 R
1
1
2
x jxj+jb xj+y
2
p
P L 2 dy ; B 2 db = jxj + jb xj + y e dy db: 1
1
1
2
This formula, and features of the lo cal time pro cess of a Brownian bridge describ ed in
the rest of this intro duction, are also implicitinRay's description of the jointlawof
x
;x 2 R andB for T an exp onentially distributed random time indep endentofthe L
T
T
Brownian motion [18, 22 , 6 ]. See [14, 17 ] for various characterizations of the lo cal time
pro cesses of Brownian bridge and Brownian excursion, and further references. These
lo cal time pro cesses arise naturally b oth in combinatorial limit theorems involving the
height pro les of trees and forests [1, 17 ] and in the study of level crossings of empirical
pro cesses [20 , 4 ].
Section 2 of this note presents an elementary derivation of formula 1 , based on
L evy's identity in distribution [15],[19, Ch. VI, Theorem 2.3 ]
d
0
L ; jB j = M ;M B where M := sup B ; 2
t t t t t s
t
0st 2
and the well known description of this jointlaw implied by re ection principle [19 , Ch.
I I I, Ex. 3.14 ], whichyields 1 for x =0.Formula 1 determines the one-dimensional
distributions of the pro cess of lo cal times at time 1 derived from a Brownian bridge of
length 1 from 0 to b, as follows: for y 0
1
2 2
x jxj+jb xj+y b
2
>yj B = b =e P L : 3
1
1
Section 3 presents a numb er of identities in distribution as consequences of this formula.
x
If x is b etween 0 and b then jxj + jb xj = jbj, so the distribution of L for the bridge from
1
0tob is the same for all x between 0 and b.Furthermore, assuming for simplicitythat
x
; 0 x b j B = b is b oth reversible and stationary. Reversibility b>0, the pro cess L
1
1
0!b
follows immediately from the fact that if B ; 0 s 1 denotes the Brownian bridge
s
of length 1 from 0 to b then
d
0!b 0!b
b B ; 0 s 1 = B ; 0 s 1 : 4
1 s s
To sp ell out the stationarity prop erty, for each x>0 and >0 with 0 there is the invariance in distribution d x+a a L ; 0 a j B = b = L ; 0 a j B = b : 5 1 1 1 1 Equivalently, for every non-negative measurable function f whichvanishes o the interval [0; ] Z Z 1 1 d 0!b 0!b f B x ds = f B ds: 6 s s 0 0 Howard and Zumbrun [11] proved 6 for f the indicator of a Borel set, and 6 for general f can b e established by their metho d. Alternatively, 6 can b e deduced from the following invariance in distribution on the path space C [0; 1]: for each0 x b d x 0!b 0!b B ; 0 s 1 = B ; 0 s 1 7 s s x 0!b 0!b where B ; 0 s 1 is derived from the bridge B ; 0 s 1 by the following s s path transformation: 8 0!b > B x if 0 s 1 x < +s x x 0!b B := s > : 0!b b B if 1 x 1 s 0!b with the rst hitting time of x by B ; 0 s 1 . The invariance 7 can b e x s checked by a standard technique [5, 20]: the corresp onding transformation on lattice 3 paths is a bijection, which gives simple random walk analogs of 7 , 6 and 5 ; the results for the Brownian bridge then followbyweak convergence. Formula 7 implies 0!b also that the pro cess ; 0 x b derived from B ; 0 s 1 has stationary x s increments.Itiseasilyshown by similar arguments that the increments of this pro cess are in fact exchangeable. According to the ab ove discussion, for each b> 0 the pro cess of bridge o ccupation times Z 1 0!b 1 0 B x ds; 0 x b s 0 has increments which are stationary and reversible, but not exchangeable due to conti- nuity of the lo cal time pro cess . See [9, 16 ] for other examples of stationary lo cal time pro cesses. In particular, it is x known [16 , Prop. 2] that if L ;t 0;x 2 R is the pro cess of lo cal times of a Brownian t x motion with drift >0, say B := B + t; t 0, then the pro cess L ;x 0 is a t t 1 x exp onentially distributed with rate . The stationarityof stationary di usion, with L 1 this pro cess is obvious by application of the strong Markov prop ertyof B at its rst hitting time of x>0. It is also known [22 , Theorem 4.5] that if T is exp onential with 1 2 rate , indep endentofB , then for each b>0 2 d B ; 0 t T j B = b = B ; 0 t 8 t T t b where is the time of the last hit of b by B , and hence b d x x ; 0 x b : 9 ; 0 x b j B = b = L L T 1 T x x The stationarityof L ;x 0 then implies stationarityof L ; 0 x b j B = b , T 1 T which is implicitinRay's description of this pro cess [18 , 22 , 6]. This yields another x pro of of the stationarityof L ; 0 x b j B = b for arbitrary t>0, by uniqueness of t t Laplace transforms. 2 Pro of of formula 1 . As observed in the intro duction, formula 1 for x = 0 is equivalent via 2 to L evy's well known description of the joint distribution of M and M B . The case of 1 with 1 1 1 x 6= 0 will now b e deduced from the case with x =0. It clearly suces to deal with x>0, as will now b e supp osed. Let := inf ft : B = xg, and set x t x B := B x t 0 : t +t x 4 x x According to the strong Markov prop ertyof B , the pro cess B := B ;t 0 is a t x x 0 x standard Brownian motion indep endentof .Let M and L b e the functionals of B x t t 0 corresp onding to the functionals M and L of B . Then for y>0, b 2 R,anda := jb xj, t t we can compute as follows: x x x 0 1 >y;B 2 db =db = P < 1 P L >y;j B j2da =da P L 1 x 1 x 1 1 x 2 x x x 1 2 da =da B >y; M P M = P < 1 1 1 1 x x x x 2 1 = P M >x+ y; M B 2 da =da 1 1 1 2 0 = P L >x+ y; B 2 da =da: 1 1 The rst and third equalities are justi ed by the strong Markovproperty, the second x app eals to L evy's identity 2 applied to B , and the fourth uses 2 applied to B . The formula 1 for x>0 can now b e read from 1 with x =0. 3 Some identities in distribution. Let R denote a random variable with the Rayleigh distribution 1 2 r 2 P R>r =e r>0 : 10 According to formula 3 , d x + L j B =0 = R 2jxj 11 1 1 x where the left side denotes the distribution of L for a standard Brownian bridge. The 1 corresp onding result for the unconditioned Brownian motion, obtained byintegrating out b in 1 , is d + x = jB j jxj : 12 L 1 1 L evy gave b oth these identities for x =0.For the bridge from 0 to b 2 R and x> 0 x the events M >x and L > 0 are a.s. identical. So 3 for y = 0 reduces to L evy's 1 1 result that 2x x b P M >xj B = b = e 0 _ b< x : 1 1 Let Z 1 1 1 2 z 2 p ' z := e ; x := ' z dz = P B >x : 1 x 2 5 Then the mean o ccupation densityat x 2 R of the Brownian bridge from 0 to b 2 R is 0 1 Z 1 jxj + jb xj x bs 1 x @ A q q ds = E L ' : 13 j B = b = 1 1 ' b 0 s 1 s s 1 s 0!b The rst equality is read from the o ccupation density formula and the fact that B s has normal distribution with mean bs and variance s 1 s . The second equality, which is not obvious directly, is obtained using the rst equalitybyintegration of 3 . As a consequence of 13 , for each b>0 and each Borel subset A of [0;b], the exp ected time sp entinA by the Brownian bridge from 0 to b is Z 1 b 0!b 14 E 1 B 2 A ds B = b = jAj 1 s ' b 0 where jAj is the Leb esgue measure of A.Take A =[0;b] to recover the standard estimate b <' b =b.For each b 2 R, the function of x app earing in 13 is the probability 0!b density function of B for U a uniform[0; 1] variable indep endent of the bridge. In U particular, for b = 0, formula 13 yields d 0!0 1 jB j = UR 15 U 2 where the Rayleigh variable R is indep endentofU . This and related identities were 0!0 found in [1], where the re ecting bridge jB j; 0 s 1 was used to describ e the s asymptotic distribution of a path derived from a random mapping. x Recall that is the rst hitting time of x by the Brownian motion B .Let denote x 1 x last time B is at x b efore time 1, with the convention = 0 if no such time, and set 1 + x x .Bywell known rst entrance and last exit decomp ositions [10], given = x 1 1 x x x B and with > 0, the segmentof B between times and is a Brownian bridge 1 x 1 1 1 x from x to x. Therefore, of length 1 q d x x R j B = b 16 L j B = b = 1 1 1 1 x x where the Rayleigh variable R is indep endentof ,and R given > 0maybeinter- 1 1 preted as the lo cal time at 0 of the standard bridge derived by Brownian scaling of the x segmentof B between times and . By consideration of moments, formula 16 shows x 1 x x that the lawof L j B = b displayed in formula 3 determines the lawof j B = b , 1 1 1 1 x and vice versa. As indicated by Imhof [12 ], the distribution of given B = b can b e 1 1 x given B = b, whichis derived byintegration from the joint distribution of and 1 x 1 6 0 easily written down. By comparison with the formula for the joint densityof and 1 jB j, due to Chung [8, 2.5 ], it turns out that 1 2 B d d 1 x x 0 j B =0; > 0 = j B =2x = 17 1 1 1 1 1 2 2 B +4x 1 where the second equality is obtained from Chung's formula by an elementary change of variable, as in [2, 6 - 8 ], where the same family of distributions on [0; 1] app ears in another context. Set a =2x and combine 11 , 16 and 17 to deduce the identity v u 2 u B d 1 t R = R a j R>a a 0 18 2 2 B + a 1 where R and B are assumed indep endent. By consideration of moments, this identity 1 amounts to the equalityoftwo di erentintegral representations for the Hermite function [13 , 10.5.2 and Problem 10.8.1]. If " is indep endentof B and exp onentially distributed with rate 1, a variation of 16 gives q d 0 0 L = R 19 2" 2" 0 where and R are indep endent, hence 2" d " = jB j R 20 1 where B and R are indep endent. By consideration of moments, this classical identity 1 is equivalent to the duplication formula for the gamma function [21 ]. Another Brownian representation of 20 is q d 0 0 j = jB 2" M 21 1 2" 2" where M is the nal value of a standard Brownian meander. Compare with [19 , Ch. 1 XI I, Ex's 3.8 and 3.9 ], and [5]. See also [3, p. 681] for an app earance of 20 in the study of random trees. Acknowledgment Thanks to Marc Yor for several stimulating conversations related to the sub ject of this note. 7 References [1] D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487{512, 1994. [2] D.J. Aldous and J. Pitman. The standard additive coalescent. Technical Rep ort 489, Dept. Statistics, U.C. Berkeley, 1997. To app ear in Ann. 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