Abstract. We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions by means of the zeros of the Bessel functions. The resulting formula is simpler and easier to treat than the corresponding one which has been already obtained.
Let {S n } be a random walk on ޚ d and let R n be the number of different points among 0, S 1 , . . . , S n−1 . We prove here that if d ≥ 2, then ψ(x) := lim n→∞ (−1/n) log P {R n ≥ nx} exists for x ≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper.We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A ⊂ ޒ d letThen φ(x) := lim t→∞ (−1/t) log P { t ≥ tx} exists for x ≥ 0 and has similar properties as ψ.
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.
Abstract. We consider the Wiener sausage for a Brownian motion up to time t associated with a closed ball in even-dimensional cases. We obtain the asymptotic expansion of the expected volume of the Wiener sausage for large t. The result says that the expansion has many log terms, which do not appear in odd-dimensional cases.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.