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BROWNIAN LOCAL TIMES LAJOS TAK,CS 1 Case Western Reserve University Department of Mathematics Cleveland, 0H 105 USA

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BROWNIAN LOCAL TIMES LAJOS TAK,CS 1 Case Western Reserve University Department of Mathematics Cleveland, 0H 105 USA Journal of Applied Mathematics and Stochastic Analysis 8, Number 3, 1995, 209-232 BROWNIAN LOCAL TIMES LAJOS TAK,CS 1 Case Western Reserve University Department of Mathematics Cleveland, 0H 105 USA (Received March, 1995; Revised June, 1995) ABSTIACT In this paper explicit formulas are given for the distribution functions and the moments of the local times of the Brownian motion, the reflecting Brown- ian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion. Key words: Distribution Functions, Moments, Local Times, Brownian Motion, Brownian Meander, Brownian Bridge, Brownian Excursion. AMS (MOS)subject classifications: 60J55, 60J65, 60J15, 60K05. 1. Introduction We consider six stochastic processes, namely, the Brownian motion, the reflecting Brown- ian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion. For each process we determine explicitly the distribution function and the moments of the local time. This paper is a sequel of the author's papers [37], [41] and [42] in which more elaborate methods were used to find the distribution and the moments of the local time for the Brownian excursion, the Brownian meander, and the reflecting Brownian motion. In this paper we shall show that we can determine all the above mentioned distributions and moments in a very simple way. We approximate each process by a suitably chosen random walk and determine the moments of the local time of the approximating random walk by making use of a conveniently chosen sequence of recurrent events. By letting the number of steps in the ran- dom walks tend to infinity we obtain the moments of local times of the processes considered. In each case the sequence of moments uniquely determines the distribution of the corresponding local time and the distribution function can be determined explicitly. It is very surprising that for each process the moments of the local time can be expressed simply by the moments of the local time of the Brownian motion. In this section we introduce the notations used and describe briefly the results obtained. The proofs are given in subsequent sections. Throughout this paper we use the following notations: 1 e- 2/2 (1) the normal density function, 1postal address: 2410 Newbury Drive, Cleveland Heights, Ohio 44118 USA. Printed in the U.S.A. ()1995 by North Atlantic Science Publishing Company 209 210 LAJOS TAK/CS x 2 (1- v_1 / u /d, the normal, distribution function, and -(- 1)xn _ Hn(x n! [1 o j!(n j)' the nth Hermite polynomial (n- O, 1,2,...). We have Hn(x xHn l(X) -(n 1)Hn 2(x) (4) for n > 2 where Ho(x 1 and Hl(x x. The jth derivative of (x) is equal to (J)(x) 1)J(x)H We define jr(a cr + 1 / e-a2x2/2(x_ 1)rdx (6) 1 for c > 0 and r- O, 1, 2, In particular, J0() v[- (7) and J(<) e </ cV[1 I:,(c)]. (8) We have Jr -I- 1() rJr- 1() Jr() (9) for r- 1, 2, The Brownian motion: Let {(t), 0 _< t _< 1} be a standard Brownian motion process. We have P{(t) <_ x} (xlV/) for 0 < t _< 1. Let us define r(a)- il measure (t:a <_ (t)< a + , 0 <_ t <_ 1} (10) for any real a. The limit (10) exists with probability one and r(a) is called the local lime al level a. The concept of local time was introduced by P. Lvy [26], [27]. See also H. Trotter [43], K. It6 and H.P. McKean, Jr. [19], and A.N. Borodin [6]. Our approach is based on a symmetric random walk {r,r > 0} where r- 1 "+" 2 q" + r for r >_ 1, 0- 0 and {tr,r _> 1} is a sequence of independent and identically distributed random variables for which P{r- 1} P{r- 1} 1/2. (11) Let us define rn(a (a- 1,2,...) as the number of subscripts r- 1,2,...,n for which (r-l-a-land (r-a" If a>_l, (r-l-a-land r-a, then we say that in the symmetric random walk a transition a- 1-a takes places at the rth step and vn(a is the number of transi- tions a- 1-a in the first n steps. In a similar way we define rn(- a) (a 1, 2,..., n) as the num- ber of subscripts r- 1,2,...,n for which (r-1 --a + 1 and (r- -a, that is, Vn(- a) is the Brownian Local Times 211 number of transitions -a + 1--,- a in the first n steps. By the results of M. Donsker [11], if uoc, the process { Jut] / V/n, 0 < t < 1} converges weakly to the Brownian motion {(t),0 < < 1}. See also I.I. Gikhman and A.V. Sorokhod [16], pp. 490-495. By using an argument of F. Knight [24], we can prove that 2'([aV/-])< nlirn P { X/r z}- P{r(a) < x} (12) for any c and x > 0. By calculating the left-hand side of (12) and letting n--,cx3 we obtain that P{-(a) < x} 2(I)( + x)- 1 (13) for x > 0. Hence it follows that 2Sr( )/V (14) for a > 0 and r > 1. We shall prove also that nlirnE V mr(a (15) for r- 1,2, The reflecting Brownian motion: We use the same notations as we used for the Brownian motion. The reflecting Brownian motion process is defined as. { [((t)[,0 < t < 1}. Its local time at level a > 0 is '(c)+ v(- a). In the same way as (12) we can prove that 2"rn([C V#])+ 2"rn(- [c lirn P { V/-])< x} P{'r(a) +'r( a) < x} (16) for a > 0 and x > 0. By calculating the moments of rn(a + rn(- a) and determining their limit behavior as noc we obtain that r 1 2 E{[r(a) + 7-( a)]r} e 1 mr((2/ 1)a) (17) for a > 0 and r _> 1 where mr(a is defined by (14). By using (17) we shall prove that P{v(a) + v( a) <_ x} 1-4 E (- 1)t-l[1-(b((2e- 1)c+x)] /=1 (18) -lxj 9(j-1) +4 E 1 (--1)(--1)J -! ((21- 1)a + x) j=i if x > 0 and a > 0. The Brownian meander: Let { + (t), 0 < _< 1} be a standard Brownian meander. The Brownian meander is a Markov process for which P{+ (0)- 0}- 1 and P{+ (t)> 0}- 1 for 0 < t < 1. If 0 < t < 1, then + (t) has a density function f(t, x). Obviously, f(t, x) 0 if x < 0. If 0 < t < 1 and x > 0, we have ,,, [2ff -1]. (19) The Brownian meander is the subject of the papers by B. Belkin [2], [3], E. Bolthausen 212 LAJOS TAK/CS [5], D.L. Iglehart [18], R.T. Durrett and D.L. Iglehart [12], R.T. Durrett, D.L. Iglehart and D.R. Miller [13], W.D. Kaigh [20], [21], [22], K.L. Chung [7] and E. Cski and S.G. Mohanty [9], [10]. Let us define the local time r + (-) for a >_ 0 by r+(a)_liml measure{t:a<+(t)<a+e, 0<t<l}. (20) The limit (20) exists with probability one and r + (a) is a nonnegative random variable. Let us define a random walk in the following way: Denote by Sn the set of sequences consisting of n elements such that each element may be either / 1 or 1 and the sum of the first elements is >_ 0 for every 1,2,..., n. The number of such sequences is ISnl- [n/21 for n- 1, 2, Let us choose a sequence at random in Sn, assuming that all the possible choices are equally probable. Denote by ( (i- 1,2,...,n) the sum of the first elements in a random sequence and set ($- 0. The sequence {(, 0 i n} describes a random walk which is usually called a Bernoulli meander. We are concerned with the random variable r (a) defined as follows: v: (a)- the number of subscripts i-1,2,...,n for which ( -a-1 and ( -a wheren>l anda>l. If n, the process { ([nt]/,O+ t 1} converges weakly to the Brownian meander {{+ (t),0 t 1} and in the same way as (12) we can prove that ([a])< + iP {2v, z}_ p{r (a) < x } (22) for any a 0 and x > 0. By calculating the moments of r (a) and determining their limit be- havior as n we obtain that E{v + (,)} 2V/[(I)(2,) (23) and m e 1 [mr- 1((21 1)-)- r 1(2-)] (24) =1 for r _> 2 where mr(. is defined by (14). By (23) and (24) we obtain that cx) k-1 l + 1 .xjr P{r (.) _< x} + 2X Z ( )kJ --! (J),2k.( + x) (J)((2k 1). + x)] (25) k=l j=0 for x > 0. The Brownian bridge: Let {ri(t),0 _< _< 1} be a standard Brownian bridge. We have P{ri(t) <_ x} ep(x/v/t(1 t)) for 0 < t < 1. Let us define w(.)- i 1. measure {t'. _< ri(t)<. + c, 0 _< t _< 1} (26) for any real .. The limit (26) exists with probability one and w(.) is a nonnegative random variable which is called the local time at level .. We consider a tied-down random walk defined in the following way" Let us denote by T2n the set of sequences consisting of 2n elements such that n elements are equal to + 1 and n ele- ments are equal to -1.
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