Asymptotic Expansions for the Expected Volume of a Stable Sausage

Abstract

Let $X_t$ be a transient stable process on $\Re^d$ and let $T_B = \inf\{t > 0: X_t \in B\}$ be the hitting time of $B$. Set $E_B(t) = \int P_x(T_B \leq t) dx$. Asymptotic expansions, as $t \rightarrow \infty$, to order 3 are obtained for all stable processes on $\Re$ that are not completely asymmetric and for all strictly stable processes on $\Re^d, d \geq 2$, whose transition density at time 1 is not zero at the origin. For those processes that are strongly transient, nontrivial $O$ estimates of the error are also obtained. Expansions to order 2 together with $O$ estimates of the error are given for the completely asymmetric processes on $\Re$, the strictly stable processes on $\Re^d$ whose transition density vanishes at 0 at time 1 and for linear Brownian motion with nonzero mean. Asymptotic expansions to order 3 together with $O$ estimates of the error are given for stable processes with drift on $\Re^d$ having exponent $\alpha < 1$. Expansions to order 3 are also given for stable processes with drift on $\Re^d$ having exponent $\alpha > 1$ when the associated drift free process is isotropic, and expansions to order 2 with $O$ estimate of the error are obtained for the other stable processes with drift on $\Re^d$ having exponent $\alpha > 1$.