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For a > 0, let W a 1 (t) and W a 2 (t) be the a-neighbourhoods of two independent standard Brownian motions in R d starting at 0 and observed until time t. We prove that, for d ≥ 3 and c > 0, lim t→∞ * Key words and phrases. Wiener sausages, intersection volume, large deviations, variational problems, Sobolev inequalities.doi:10.4007/annals.2004.159.741 fatcat:wv7id3kdubbqfiar6i4n46mfp4