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Hitting times to spheres of Brownian motions with drifts starting from the origin.

1. Introduction. This article deals with the first passage problem of a Brownian motion with a constant drift. Let [mathematical expression not reproducible] be a standard Brownian motion on [R.sup.d] starting from a given point x 2 [R.sup.d]. For a constant vector % 2 [R.sup.d] a Brownian motion with a drift [??], denoted by [mathematical expression not reproducible], is defined as

[mathematical expression not reproducible],

which implies the first hitting time of [mathematical expression not reproducible] to the sphere [S.sup.d-1.sub.r] with radius r and centered at the origin.

In this paper we will discuss the probability density function of [[tau].sub.r.sup.([??])], for which we write [p.sub.r.sup.([??])] (.; x) in the case when d [greater than or equal to] 2. Explicit forms of [p.sub.r.sup.(0)]) (.; x) are obtained in [1,6,7] for [absolute value of x] < r and in [4] for [absolute value of x] > r, where [absolute value of y] is the Euclidean distance between [absolute value of x] 2 [R.sup.d] and the origin. In the case [??] [not equal to] 0, formulas for [p.sub.r.sup.([??]) (.; x) have been deduced for x [not equal to] 0. One of the formulas is given in [5, Theorem 1.1] and expressed as an infinite sum of which each summand consists of the modified Bessel functions, the Gegenbauer polynomials and the densities [p.sub.r.sup(0)] (.;x). Other form is represented in [11] by an integral involving the Bessel functions. We should remark that a general framework for discussing the distribution of the hitting time is provided in [10].

One of our purposes of this paper is to give an explicit form of [p.sub.r.sup.([??])] (.; 0) when [??] [not equal to] 0. For simplicity we use the notation [p.sub.r.sup.([??])] (*) instead of [p.sub.r.sup.([??])]) (.;0). We obtain that [p.sub.r.sup.([??])] is represented by the density [p.sub.r.sup.0] and the modified Bessel function [I.sub.u] of the first kind of order [mu]. For convenience we put v = d/2 - 1.

Theorem 1.1. Let d [??] 2 and [??] [not equal to] 0. We have that

[mathematical expression not reproducible]

for any t > 0.

The idea of the proof is to represent the Laplace transform of [[tau].sup.([??]).sub.r] as an integral with respect to the distribution of ([tau], [B.sub.[tau]]) by the Cameron-Martin formula, which is similar to the calculation used in [5]. Here the notation [tau] has been used instead of [[tau].sup.0.sub.r] for simplicity. A proof of Theorem 1.1 will be given in the next section. We should mention that the formula for [pr.sup.([??])] (t; 0) can not be simply deduced by taking a limit of [pr.sup.([??])] (t; x), given in [5, Theorem 1.1], as x tends to 0 since the formula for [pr.sup.([??])] (t; x) has terms which contain ([??], x)/([absolute value of [??]] [absolute value of x]), where (%,x) is the standard inner product of [??] and x. In addition, we remark that the explicit form of [pr.sup.(0)] is provided in the following way:

(1:1) [mathematical expression not reproducible]

where [J.sub.[mu]] the Bessel function of the first kind of order [mu] and {[j.sub.[mu],n}.sup.[infinity].sub.n=1] 1 is the increasing sequence of positive zeros of [J.sub.[mu]] (cf. [1, Theorem 2]).

Another purpose of this paper is to give the asymptotic behavior of the tail probability of [[tau]r.sup([??])] for [??] [not equal to] 0. The following theorem can be deduced from (1.1) and Theorem 1.1.

Theorem 1.2. Let d [??] 2 and [??] [not equal to] 0. We have that

[mathematical expression not reproducible]

as t [right arrow] 1.

We will prove the theorem in Section 3.

2. The density function. In this section we give a proof of Theorem 1.1 with the help of the Laplace transform of [[tau]r.sup.([??])]. When x [not equal to] 0, the Laplace transform of [[tau]r.sup.([??])] is represented in [5, p. 5391]. In the same way we can deduce that

[mathematical expression not reproducible]

for x = 0 and the right-hand side is equal to

(2.1) [mathematical expression not reproducible]

We omit the detailed calculation. It is known that

[mathematical expression not reproducible] for t [??] 0 and a Borel set A in [S.sup.d-1.sub.r], where the notation [[sigma].sub.r] has been used to denote the uniform distribution on [S.sup.d-1.sub.r], (cf. [8, p. 27]). This implies that (2.1) can be represented by

(2.2) [mathematical expression not reproducible].

Hence it is sufficient to calculate the integral on y in (2.2). We have that

(2.3) [mathematical expression not reproducible].

Let w = (1,0, ..., 0) [member of] [R.sup.d]. We take an orthogonal matrix T of order d such that T[??] = [absolute value of [??]] w. Since [[sigma].sub.1] is preserved under orthogonal linear transformations on [R.sup.d], the right-hand side of (2.3) is equal to

(2.4) [mathematical expression not reproducible].

It is easy to see that, if [??] = 3, the right-hand side of

(2.4) coincides with the product of the following two integrals:

(2.5) [mathematical expression not reproducible],

(2.6) [mathematical expression not reproducible],

where [S.sub.d-1] is used for the surface area of [S.sup.d-1.sub.1]. We can find that (2.5) coincides with

[square root of([pi] [(2/r[absolute value of [??]).sup.v] [GAMMA] (v + 1/2) [I.sub.v] ([r[absolute value of [??])]

in [3,p. 491] and it is obvious that the integral in (2.6) is equal to [S.sub.d-2.] By the well-known formula [S.sub.m-1] = [2[pi].sup.m/2]/[GAMMA](m/2) for each m [??] 2, we obtain that the right-hand side of (2.4) and also (2.3) are equal to

[2.sup.v][GAMMA] (v +1). (r[I.sub.v]([absolute value of [??]]/ ([r[I.sub.v]([absolute value of [??]].sup.v].

This yields that (2.2), which is the Laplace transform of [[tau]r.sup.[??]]), can be expressed by

(2.7) [mathematical expression not reproducible]

in the case d [??] 3.

Note that v = 0 if d = 2. The calculation in the two dimensional case is easy. Indeed, we have that the right-hand side of (2.4) is

[mathematical expression not reproducible]

(cf. [3, p. 491]). This implies that (2.2) has the same form as (2.7) for v = 0.

We complete the proof of Theorem 1.1.

3. Asymptotics of the tail probability.

This section is devoted to showing Theorem 1.2.

Theorem 1.1 gives that

(3.1) [mathematical expression not reproducible]

In addition, it is known that

[mathematical expression not reproducible]

for [lambda] > 0 (cf. [2,6]). Thus we immediately conclude that the right-hand side of (3.1) is equal to 1, which implies that P([[tau]r.sup.([??])] < 1) = 1. By Theorem 1.1 and (1.1) we have that P[[tau]r.sup.([??])] > t) is equal to

(3.2) [mathematical expression not reproducible]

In order to prove Theorem 1.2, we should justify changing the order of summation and integration in (3.2).

It is well-known that

(3.3) [j.sub.v,n] = [pi]n + O[1]

for large n (cf. [9, p. 506]). Moreover, combining (3.3) and the asymptotic behavior of the Bessel function of the first kind (cf. [9, p. 199]), we can obtain that

(3.4) [J.sub.v+1] ([j.sub.v,n]) = (-1).sup.n+1] [pi]/[square root of 2] [square root of n]

as n [right arrow] [infinity] (cf. [7, p. 318]). It is easy to show by (3.3) and (3.4) that

[mathematical expression not reproducible]

which implies that there exists a constant C such that

(3.5) [mathematical expression not reproducible]

for each n [??] 1. Hence we deduce from (3.3) and (3.5) that

[mathematical expression not reproducible]

converges for each t > 0. We can change the order of the summation and the integral in (3.2) and thus it follows that P([[tau]r.sup.([??]0] > t) is equal to

[mathematical expression not reproducible]

Using (3.5) again, we obtain the claim of Theorem 1.2.

Acknowledgment. This work is partially supported by the Grant-in-Aid for Scientific Research (C) No. 16K05208 of Japan Society for the Promotion of Science (JSPS).


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doi: 10.3792/pjaa.95.37

Dedicated to Professor Hiroyuki Matsumoto on the occasion of his 60th birthday


Department of Mathematics, Kumamoto University, 2-39-1 Kurokami, Chuo-ku, Kumamoto 860-8555, Japan (Communicated by Masaki KASHIWARA, M.J.A., March 12, 2019)
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Author:Hamana, Yuji
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
Article Type:Report
Date:Apr 1, 2019
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