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

Abstract

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.

The Transition Density of Brownian Motion Killed on a Bounded Set

“…Here | · | designates the Lebesgue measure on R d and ∂A the Euclidean boundary of A. The asymptotic form of the lateral part is computed in [12] and our present result together with it identifies the explicit asymptotic form of the caloric measure for large time valid uniformly at least in the regime |x| ∨ |y| = o(t), where x ∨ y = max{x, y}.…”

“…Let d = 2 and e A (x) be the Green function for Ω A with a pole at infinity normalized so that e A (x) ∼ lg x when x → ∞ (see (12) for a definition). The following theorem is stated only in the case x ≤ y which is not a restriction because p A t (x, y) is symmetric in x and y as mentioned above.…”

“…Relations (7), (8) and (9) are taken from [9], [11] and [12], respectively. Roughly speaking we first obtain (4), by using (7) and (8), under the condition x ∼ y ∼ √ t/ lg t (a special case of the first one of (4)).…”

“…But according to Theorem A.1 we have q Proof of Proposition 2.3. In [12] it is shown that there exists a measure λ A v (dξ) on ∂A such that λ A v (dξ) depends on v continuously, the total measure λ A v (∂A) is positive, and H A (y, t; ·)…”

“…Here we collect several results from [10], [11] and [12], the original theorems which they are taken or adapted from being indicated in the square brackets.…”