The Transition Density of Brownian Motion Killed on a Bounded Set

2016

Potential Anal

Self Cite

running head: random walk killed on a finite set key words: two-dimensional random walk; exterior domain; transition probability; overshoot estimates AMS Subject classification (2010): Primary 60G50, Secondary 60J45. AbstractWe study asymptotic behavior, for large time n, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset A. We show that it behaves like 4ũ A (x)ũ −A (−y)(lg n) −2 p n (y − x) for large n, uniformly in the parabolic regime |x| ∨ |y| = O( √ n), where p n (y − x) is the transition kernel of the random walk (without killing) andũ A is the unique harmonic function in the 'exterior of A' satisfying the boundary conditionũ A (x) ∼ lg |x| at infinity.

“…By (12) and (24) we also obtain that κu A (x) ∼ lg |x| (|x| → ∞) as stated in (5). Now we prove Proposition 1, which we state again…”

Section: The Function U a And Proof Of Propositionsupporting

confidence: 63%

“…For the description of the strategy of the proof of Theorem 1 the readers are referred to [12]: in the latter half of its first section the skeleton of proof to the corresponding result for Brownian motion is given, by which the role of Proposition 1 may be fully understood.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic Behaviour of a Random Walk Killed on a Finite Set

2016

Potential Anal

Self Cite

“…, representing the same probability but for the dual walk started at −∞. Thus g − −A (−ξ) is the probability that the dual walk 'started at −∞' hits −A at −ξ, whence 14) and similarly g + −A (−ξ) = H −∞ A (ξ). By (1.14) and by (1.11) we especially have…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…The same method is applied in [13] to higher dimensional random walks to obtain a similar strengthening of Kesten's result. For multidimensional Brownian motions the corresponding problem is studied by [3] for space variables restricted to compact sets and by [14] without any restriction as such.…”

mentioning

confidence: 99%

One dimensional random walks killed on a finite set

Stochastic Processes and their Applications

2017

Self Cite

“…If

A

is compact and

Ω_{A}

denotes the unbounded component of

R^{d} ∖ A

, then

Q_{A} false(boldx, d t d ξ false)

is identified with the lateral part of the caloric measure (or parabolic measure) for the heat operator

\frac{1}{2} Δ - \partial_{t}

in the space‐time domain

D = {false(boldx, t false) \in R^{d} \times false(0, \infty false) : x \in {normalΩ}_{A}}

, the exterior of a cylinder (see Section A.1). The other part of it is nothing but the measure whose density is given by the heat kernel for

Ω_{A}

with Dirichlet zero boundary condition and its (uniform) asymptotic estimate for large time is recently obtained by for the space variables restricted to any compact set and by without restriction. The present work is partly motivated and steered by a study of Wiener sausage swept by the set

A

attached to a

d

‐dimensional Brownian motion started at the origin.…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

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

Proc. London Math. Soc.

2017

Self Cite

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.