On the expected volume of the Wiener sausage

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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“…We are ready to prove Proposition 2.2. Similarly to Proposition 3.1 in [5], we can show that, for λ > 0…”

Section: Laplace Transform Of Lsupporting

confidence: 63%

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

Hamana

Matsumoto

2016

Forum Mathematicum

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“…Moreover, the function (K d/2 /K d/2−1 )(r √ 2λ) in λ > 0 represents the Laplace transform of the expectation of the Wiener sausage for the d-dimensional Brownian motion associated with a close ball with radius r (cf. [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 .…”

Section: Introductionmentioning

confidence: 99%

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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“…For details, see [5,10,14]. In higher-dimensional cases, although the explicit form of P x [τ t] has been obtained in [8], it seems to be difficult to carry out the integration on x.…”

Section: It Is Easy To See That the Expectation Of The Volume Of W (Tmentioning

confidence: 99%

“…The central limit theorems are proved in [13] and the results concerning large deviations are given in [1,2,7]. This paper deals with the case that the non-polar compact set is a closed ball with radius r. If the dimension is odd, the explicit form of the mean volume of the Wiener sausage and its asymptotic expansion have been obtained in [5,6]. They are represented by zeros of a suitable modified Bessel function.…”

Section: Introductionmentioning

confidence: 99%