The Distribution of Local Times of a Brownian Bridge
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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 [email protected] 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 xL 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+jbxj+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 RandB 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], whichyields1for 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+jbxj+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 1s s To sp ell out the stationarity prop erty, for each x>0 and >0 with 0 <x+ b, 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 xds = 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 <s 1 x 1s 0!b with the rst hitting time of x byB ; 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 10 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 stationarityofL ;x 0 then implies stationarityofL ; 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 stationarityofL ; 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 jjxj : 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 >xandL > 0 are a.s. identical. So 3 for y = 0 reduces to L evy's 1 1 result that 2xxb 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 s1 s s1 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 s1 s.