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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).

References

[1] Z. Ciesielski and S. J. Taylor, First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103 (1962), 434 450.

[2] R. K. Getoor and M. J. Sharpe, Excursions of Brownian motion and Bessel processes, Z. Wahrsch. Verw. Gebiete 47 (1979), no. 1, 83 106.

[3] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Academic Press, Amsterdam, 2007.

[4] Y. Hamana and H. Matsumoto, The probability densities of the first hitting times of Bessel processes, J. Math-for-Ind. 4B (2012), 91 95.

[5] Y. Hamana and H. Matsumoto, Hitting times to spheres of Brownian motions with and without drifts, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5385 5396.

[6] J. Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), no. 5, 760 770.

[7] J. T. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 309 319.

[8] S. C. Port and C. J. Stone, Brownian motion and classical potential theory, Academic Press, New York, 1978.

[9] G. N. Watson, A treatise on the theory of Bessel functions, reprint of the 2nd (1944) ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.

[10] M. Yamazato, Hitting time distributions of single points for 1-dimensional generalized diffusion processes, Nagoya Math. J. 119 (1990), 143-172.

[11] C. Yin and C. Wang, Hitting time and place of Brownian motion with drift, Open Stat. Prob. J. 1 (2009), 38 42.

doi: 10.3792/pjaa.95.37

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

By Yuji HAMANA

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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