Asymptotic Expansion of the Expected Volume of the Wiener Sausage in Even Dimensions

Hamana, Yuji

doi:10.2206/kyushujm.70.167

Cited by 8 publications

(13 citation statements)

References 15 publications

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“…Moreover (2.4) has been already established in [6]. Therefore we concentrate on the case when ν − 1/2 is not an integer.…”

Section: Ratios Of Macdonald Functionsmentioning

confidence: 94%

“…Hence the Laplace transform on (4.1) can be inverted. When d is odd and more than or equal to five, Theorem 1.1 in [6] shows that, for t > 0…”

Section: The Expected Volume Of the Wiener Sausagementioning

confidence: 99%

“…[5]). In the case when d is odd, Hamana [6] divided the function into the sum of several functions of which the inverse Laplace transforms can be obtained easily and deduced an exact form of the mean volume of the Wiener sausage by means of zeros of K d/2−1 . By using our result we can show that, also in the even dimensional case, the expectation is represented in a similar form.…”

Section: Introductionmentioning

confidence: 99%

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Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions

Hamana¹,

Matsumoto

2016

J. Math. Soc. Japan

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We derive formulae for some ratios of the Macdonald functions, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the Lévy measure of the distribution of the first hitting time of a Bessel process and the other is an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. Moreover, the result enables us to write down the algebraic equations whose roots are the zeros of Macdonald functions.

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“…Moreover (2.4) has been already established in [6]. Therefore we concentrate on the case when ν − 1/2 is not an integer.…”

Section: Ratios Of Macdonald Functionsmentioning

confidence: 94%

“…Hence the Laplace transform on (4.1) can be inverted. When d is odd and more than or equal to five, Theorem 1.1 in [6] shows that, for t > 0…”

Section: The Expected Volume Of the Wiener Sausagementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions

Hamana¹,

Matsumoto

2016

J. Math. Soc. Japan

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“…We note that the equality (2.11) is already known [4,Lemma 3.2] and it is a corresponding Mittag-Leffler expansion for K ν . Now, let us notice that from (2.11) it follows that n k=1,k =j 1 z ν,k − z ν,j = lim…”

mentioning

confidence: 63%

On an identity for zeros of Bessel functions

Baricz

Maširević

Pogány

et al. 2015

Journal of Mathematical Analysis and Applications

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Abstract. In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.

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“…which coincides with the Newtonian capacity of D. Moreover the several smaller terms of L (d) 0 (t) are given in [6,7,10,17,18]. We consider the same problem in the case of v = 0 and put…”

Section: Large Time Asymptoticsmentioning

confidence: 99%

A formula for the expected volume of the Wiener sausage with constant drift

Hamana

Matsumoto

2016

Forum Mathematicum

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We consider the Wiener sausage for a Brownian motion with a constant drift up to time t associated with a closed ball. In the two or more dimensional cases, we obtain the explicit form of the expected volume of the Wiener sausage. The result says that it can be represented by the sum of the mean volumes of the multi-dimensional Wiener sausages without a drift. In addition, we show that the leading term of the expected volume of the Wiener sausage is written as κt(1+o [1]) for large t and an explicit form gives the constant κ. The expression is of a complicated form, but it converges to the known constant as the drift tends to 0.2010 Mathematics Subject Classification. Primary 60J65; Secondary 44A10.

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Asymptotic Expansion of the Expected Volume of the Wiener Sausage in Even Dimensions

Cited by 8 publications

References 15 publications

Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions

Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions

On an identity for zeros of Bessel functions

A formula for the expected volume of the Wiener sausage with constant drift

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