Density of Space-Time Distribution of Brownian First Hitting of a Disc and a Ball

Uchiyama, Kôhei

doi:10.1007/s11118-015-9516-z

Cited by 13 publications

(21 citation statements)

References 12 publications

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“…The following theorem, applied in Section 4.2.3, follows from Corollary 2.1 and Lemma 5.6 of [27] ((i) is immediate from the former; as for (ii) use the latter in addition).…”

Section: )mentioning

confidence: 98%

“…Here we consider the case when

A = U (a)

, the open ball centered at the origin of radius

a

, and state the asymptotic estimates of

H_{U (a)} false(boldx, t; d ξ false)

obtained in . The following notation is used throughout the paper.…”

Section: The Hitting Distribution For a Disc/ballmentioning

confidence: 99%

“…Next we present a result from on the conditional distribution of the hitting site

B_{σ false(U false(a false) false)}

given

σ_{U (a)} = t

. For

α ⩾ 0

let

g_{α} false(θ false)

be a function of

θ \in [- π, π]

given by

0true g_{α} (θ) = \sum_{n = 0}^{\infty} \frac{K_{0} (α)}{K_{n} (α)} H_{n} (θ)

α > 0

and

g_{0} false(θ false) \equiv 1

, where

H_{n} false(θ false) = \{\begin{matrix} left prefixcos n θ & left if d = 2, \\ left κ_{n, ν} C_{n}^{ν} (cos θ) & left if d ⩾ 3 . \end{matrix}

Here

κ_{n, ν} = false(n + ν false) Γ false(ν false) / \sqrt{π} Γ false(ν + \frac{1}{2} false)

and

C_{n}^{ν} false(z false)

is the Gegenbauer polynomial of order

n

associated with

ν

…”

Section: The Hitting Distribution For a Disc/ballmentioning

confidence: 99%

“…In the Huygens representation of

H_{A} false(boldx, t; E false)

we restrict the range of integration to

[0, η] \times {ξ \in \partial U (R) : x \cdot ξ > 0}

and apply Theorem to see

0true H_{A} (x, t; E) ⩾ R^{2 ν} \int_{0}^{η} d s \int_{ξ \cdot boldx > 0} \frac{ξ \cdot boldx}{t - s} p_{t - s}^{false(d false)} (ξ - x) H_{A} (ξ, s; E) m_{R} (d ξ) (1 + o false(1 false)),

where

o (1) \to 0

x \to \infty

(cf. [, Lemma 5.1] if necessary). In the integrand we may replace

t - s

t - η

for the lower bound and integration of

H_{A} false(ξ, s; E false)

over

0 < s < η

yields

P_{ξ} false[B_{σ false(A false)} \in E, σ_{A} ⩽ η false]

.…”

Section: The Hitting Distribution For a Bounded Borel Setmentioning

confidence: 99%

“…Next we present a result from [27] on the conditional distribution of the hitting site B σ(U (a)) given σ U (a) = t. For α 0 let g α (θ) be a function of θ ∈ [−π, π] given by…”

Section: )mentioning

confidence: 99%

See 4 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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“…The following theorem, applied in Section 4.2.3, follows from Corollary 2.1 and Lemma 5.6 of [27] ((i) is immediate from the former; as for (ii) use the latter in addition).…”

Section: )mentioning

confidence: 98%

“…Here we consider the case when

A = U (a)

, the open ball centered at the origin of radius

a

, and state the asymptotic estimates of

H_{U (a)} false(boldx, t; d ξ false)

obtained in . The following notation is used throughout the paper.…”

Section: The Hitting Distribution For a Disc/ballmentioning

confidence: 99%

“…Next we present a result from on the conditional distribution of the hitting site

B_{σ false(U false(a false) false)}

given

σ_{U (a)} = t

. For

α ⩾ 0

let

g_{α} false(θ false)

be a function of

θ \in [- π, π]

given by

0true g_{α} (θ) = \sum_{n = 0}^{\infty} \frac{K_{0} (α)}{K_{n} (α)} H_{n} (θ)

α > 0

and

g_{0} false(θ false) \equiv 1

, where

H_{n} false(θ false) = \{\begin{matrix} left prefixcos n θ & left if d = 2, \\ left κ_{n, ν} C_{n}^{ν} (cos θ) & left if d ⩾ 3 . \end{matrix}

Here

κ_{n, ν} = false(n + ν false) Γ false(ν false) / \sqrt{π} Γ false(ν + \frac{1}{2} false)

and

C_{n}^{ν} false(z false)

is the Gegenbauer polynomial of order

n

associated with

ν

…”

Section: The Hitting Distribution For a Disc/ballmentioning

confidence: 99%

“…In the Huygens representation of

H_{A} false(boldx, t; E false)

we restrict the range of integration to

[0, η] \times {ξ \in \partial U (R) : x \cdot ξ > 0}

and apply Theorem to see

0true H_{A} (x, t; E) ⩾ R^{2 ν} \int_{0}^{η} d s \int_{ξ \cdot boldx > 0} \frac{ξ \cdot boldx}{t - s} p_{t - s}^{false(d false)} (ξ - x) H_{A} (ξ, s; E) m_{R} (d ξ) (1 + o false(1 false)),

where

o (1) \to 0

x \to \infty

(cf. [, Lemma 5.1] if necessary). In the integrand we may replace

t - s

t - η

for the lower bound and integration of

H_{A} false(ξ, s; E false)

over

0 < s < η

yields

P_{ξ} false[B_{σ false(A false)} \in E, σ_{A} ⩽ η false]

.…”

Section: The Hitting Distribution For a Bounded Borel Setmentioning

confidence: 99%

“…Next we present a result from [27] on the conditional distribution of the hitting site B σ(U (a)) given σ U (a) = t. For α 0 let g α (θ) be a function of θ ∈ [−π, π] given by…”

Section: )mentioning

confidence: 99%

See 3 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.

Self Cite

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show abstract

Asymptotic behaviour of the Bessel heat kernels

Bogus

2019

Math. Z.

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The Transition Density of Brownian Motion Killed on a Bounded Set

Uchiyama

2017

J Theor Probab

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We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set A. We derive the asymptotic form of the density, say p A t (x, y), for large times t and for x and y in the exterior of A valid uniformly under the constraint |x|∨|y|is the transition kernel of the Brownian motion (without killing) and e A is the Green function for the 'exterior of A' with a pole at infinity normalized so that e A (x) ∼ lg |x|. We also provide fairly accurate upper and lower bounds of p A t (x, y) for the case |x| ∨ |y| > t as well as corresponding results for the higher dimensions.

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Density of Space-Time Distribution of Brownian First Hitting of a Disc and a Ball

Cited by 13 publications

References 12 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

Asymptotic behaviour of the Bessel heat kernels

The Transition Density of Brownian Motion Killed on a Bounded Set

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