Moderate deviations for the volume of the Wiener sausage

Abstract

For a>0,let W^a(t) be the a-neighbourhood of standard Brownian motion in R^d starting at 0 and observed until time t.It is well-known that E|W^a(t)|~kappa_a t (t->infty) for d >= 3,with kappa_a the Newtonian capacity of the ball with radius a. We prove that lim_{t->infty} 1/t^{(d-2)/d}log P(|W^a(t)|<=bt) = -I^{kappa_a}(b) in (-infty,0) for all 0<b<kappa_a and derive a variational representation for the rate function I^{kappa_a}.We show that the optimal strategy to realise the above moderate deviation is for W^a(t) to look like a Swiss cheese: W^a(t) has random holes whose sizes are of order 1 and whose density varies on scale t^{1/d}.The optimal strategy is such that t^-1/d W^a(t) is delocalised in the limit as t->infty.This is markedly different from the optimal strategy for large deviations |W^a(t)|<=f(t) with f(t)=o(t),where W^a(t) is known to fill completely a ball of volume f(t) and nothing outside,so that W^a(t) has no holes and f(t)^{-1/d}W^a(t) is localised in the limit as t->infty.We give a detailed analysis of the rate function I^{kappa_a},in particular,its behaviour near the boundary points of (0,kappa_a).It turns out that I^{kappa_a} has an infinite slope at kappa_a and,remarkably,for d>=5 is nonanalytic at some critical point in (0,kappa_a),above which it follows a pure power law.This crossover is associated with a collapse transition in the optimal strategy.We also derive the analogous moderate deviation result for d=2.In this case E|W^a(t)|~2pi t/log t (t->infty),and we prove that lim_{t->infty} 1/log t log P(|W^a(t)|<=bt/log t) =-I^{2pi}(b)in (-infty,0) for all 0<b<2pi.