The expected area of the Wiener sausage swept by a disc

Uchiyama, Kôhei

doi:10.1016/j.spa.2012.09.005

Cited by 5 publications

(7 citation statements)

References 23 publications

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“…Fine estimates are obtained in the case when the process is pinned at the origin by McGillivray (

d ⩾ 3

) and (

d = 2

) (cf. also ) and in the case when

d = 2

, the value at time

t

is pinned within a parabolic region and

A

is a disc by . There are many works for the sausage in the unconditional case (see, for example, the references in ).…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

“…Proof. The proof is similar to that of Lemma 4.3 of [25]. Let h > 4R 2 K be a constant that will be suitably chosen later depending on M (see the part (c) below).…”

Section: The Wiener Sausage For a Brownian Bridgementioning

confidence: 92%

“…Remark (a)In the case when

B_{t}

is pinned at

x = 0

an asymptotic expansion of

E_{bold0} false[{vol}_{d} (S_{K} false(t false)) false| B_{t} = bold0 false]

is obtained in (

d ⩾ 3

) and (

d = 2

) ( see also ). (b)In , it is shown that if

d = 2

, for each

M > 1

, uniformly for

false| boldx false| ⩽ M \sqrt{t}

E_{bold0} false[area (S_{U (a)} false(t false)) false| B_{t} = boldx false] = 2 π t N false(κ t / a^{2} false) + \frac{π x^{2}}{{false(prefixlg t false)}^{2}} true[prefixlg \frac{t}{x^{2} \lor 1} + O (1) true] + O false(1 false)

t \to \infty

, where

κ = 2 e^{- 2 γ}

and

0true N (λ) = \int_{0}^{\infty} e^{- λ u} false[{(lg u)}^{2} + π^{2} false)

…”

Section: The Wiener Sausage For a Brownian Bridgementioning

confidence: 94%

See 2 more Smart Citations

The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge

Uchiyama

2017

Proc. London Math. Soc.

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The space‐time distribution, QAfalse(boldx,dt0.16emdξfalse) say, of Brownian hitting of a bounded Borel set A of boldRd is studied. We derive the asymptotic form of the leading term of the time‐derivative QAfalse(boldx,dt0.16emdξfalse)/dt for each d=2,3,…, valid uniformly with respect to the starting point boldx of the Brownian motion, which result significantly extends the classical ones for QAfalse(boldx,dt0.16emdξfalse) itself by Hunt (d=2), Joffe and Spitzer (d⩾3). The results obtained are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point.

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“…Fine estimates are obtained in the case when the process is pinned at the origin by McGillivray (

d ⩾ 3

) and (

d = 2

) (cf. also ) and in the case when

d = 2

, the value at time

t

is pinned within a parabolic region and

A

is a disc by . There are many works for the sausage in the unconditional case (see, for example, the references in ).…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

“…Proof. The proof is similar to that of Lemma 4.3 of [25]. Let h > 4R 2 K be a constant that will be suitably chosen later depending on M (see the part (c) below).…”

Section: The Wiener Sausage For a Brownian Bridgementioning

confidence: 92%

“…Remark (a)In the case when

B_{t}

is pinned at

x = 0

an asymptotic expansion of

E_{bold0} false[{vol}_{d} (S_{K} false(t false)) false| B_{t} = bold0 false]

is obtained in (

d ⩾ 3

) and (

d = 2

) ( see also ). (b)In , it is shown that if

d = 2

, for each

M > 1

, uniformly for

false| boldx false| ⩽ M \sqrt{t}

E_{bold0} false[area (S_{U (a)} false(t false)) false| B_{t} = boldx false] = 2 π t N false(κ t / a^{2} false) + \frac{π x^{2}}{{false(prefixlg t false)}^{2}} true[prefixlg \frac{t}{x^{2} \lor 1} + O (1) true] + O false(1 false)

t \to \infty

, where

κ = 2 e^{- 2 γ}

and

0true N (λ) = \int_{0}^{\infty} e^{- λ u} false[{(lg u)}^{2} + π^{2} false)

…”

Section: The Wiener Sausage For a Brownian Bridgementioning

confidence: 94%

See 1 more Smart Citation

The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge

Uchiyama

2017

Proc. London Math. Soc.

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“…Fine estimates are obtained in the case when the process is pinned at the origin by McGillivray [18] (d ≥ 3) and [19] (d = 2) (cf. also [3]) and in the case when d = 2, the value at time t is pinned within a parabolic region and A is a disc by [25]. There are many works for the sausage in the unconditional case (see, e.g., the references of [3]).…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

Brownian hitting distributions in space-time of bounded sets and the expected volume of Wiener sausage for Brownian bridges

Uchiyama

2014

Preprint

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The space-time distribution, Q A (x, dtdξ) say, of Brownian hitting of a bounded Borel set A of R d is studied. We derive the asymptotic form of the leading term of the timederivative Q A (x, dtdξ)/dt for each d = 2, 3, . . ., valid uniformly with respect to the starting point x of the Brownian motion, which result significantly extends the classical ones for Q A (x, dtdξ) itself by Hunt (d = 2), Joffe and Spitzer (d ≥ 3). The results obtained are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point.

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“…, an analogue to the explicit form of q (3) , can take the place of the leading term in the formula (19). Since the difference of them is at most the magnitude of O(x (2−2ν)∨1 /t ν+2 ), this is true if ν < 1; in the case ν > 1, however, the difference becomes much larger than η(t, x) as x gets large, so that the replacement causes a larger error term.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Asymptotics of the densities of the first passage time distributions for Bessel diffusions

Uchiyama¹

2014

Trans. Amer. Math. Soc.

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This paper concerns the first passage times to a point a > 0, denoted by σ a , of Bessel processes. We are interested in the case when the process starts at x > a and compute the densities of the distributions of σ a to obtain the exact asymptotic forms of them as t → ∞ that are valid uniformly in x > a for every order of Bessel process. 1 Introduction and main resultsThis paper concerns the first passage times to a point a ≥ 0, denoted by σ a , of Bessel processes of order ν ∈ R. We are interested in the case when the process starts at x > a and compute the densities of the distributions of σ a to obtain the exact asymptotic forms of them as t → ∞ that are valid uniformly in x > a for each order ν. If ν = ±1/2, we have well-known explicit expressions of them, which are often used in various circumstances, while otherwise there has been quite restricted information on them until quite recently. In the case when 0 ≤ x < a the distribution of σ a solves a boundary value problem of the associated second order differential equation on the finite interval (0, a) and the distribution of σ a or its density is represented by means of eigenfunction expansion ([3], [8], [12], etc.) and thereby we can obtain accurate estimates of them. In the case x > a, however, the region for the differential equation is the infinite interval (a, ∞) and the corresponding representation is given by a Fourier-Bessel transform (cf.[16]: Section 4.10), which it seems not simple a matter to derive an asymptotic form of the density directly from and there have been only a few partial results as given in [15], [18], [7] in which ν = 0 or/and relative ranges of x are restricted at least for sharp estimates (in addition to the cases ν = ±1/2). In the recent paper [2] Byczkowski, Malecki and Ryznar have computed an estimate of the density for σ a for all values of ν by using a certain integral representation of it given in [1]: they obtain upper and lower bounds of the correct order of magnitude valid uniformly for all t > 0, x > a, which however does not give the exact asymptotic form as we shall obtain in this paper (although in some cases their results are very close to and even finer than ours, see (i) of Remark 1 of the present paper). Hamana and Matumoto [10] have derived a similar (but, in a significant point, quite different) integral representation of the density of σ a for the case x > a (as 1 key words: first passage time, exterior problem, uniform estimate, Bessel diffusion

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The expected area of the Wiener sausage swept by a disc

Cited by 5 publications

References 23 publications

The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge

The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge

Brownian hitting distributions in space-time of bounded sets and the expected volume of Wiener sausage for Brownian bridges

Asymptotics of the densities of the first passage time distributions for Bessel diffusions

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