A large-deviation result for the range of random walk and for the Wiener sausage

Let {S-n} be a random walk on Z(d) and let R-n be the number of different p oints among 0, S-1.....Sn-1. We prove here that if d greater than or equal to 2. then psi (x) := lim(n --> proportional to)(-1/n) log P {R-n greater t han or equal to nx} exists for x greater than or equal to 0 and establish s ome convexity and monotonicity properties of psi (x). The one-dimensional c ase will be treated in a separate paper. We also prove a similar result fur the Wiener sausage (with drift). Let B(t ) be a d-dimensional Brownian motion with constant drift, and for a bounded set A subset of R-d let Lambda (t) = Lambda (t)(A) be the d-dimensional Le besgue measure of the 'sausage' U-o less than or equal tos less than or equ al tot (B(s) + A) Then phi (x) := lim (t --> proportional to) (-1/t) log P{ A(t)greater than or equal to tx} exists for x greater than or equal to0 and has similar properties as psi.