TY - JOUR
T1 - On the volume of the intersection of two Wiener sausages
AU - Berg, van den, M.
AU - Bolthausen, E.
AU - Hollander, den, W.Th.F.
PY - 2004
Y1 - 2004
N2 - 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¿8 1 t (d-2)/d logP(|W a 1 (ct)nW a 2 (ct)|=t)=-I ¿ a d (c) and derive a variational representation for the rate constant I ¿ a d (c) . Here, ¿ a is the Newtonian capacity of the ball with radius a . We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to "form a Swiss cheese": the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1/d according to a certain optimal profile.
We study in detail the function c¿I ¿ a d (c) . It turns out that I ¿ a d (c)=T d (¿ a c)/¿ a , where T d has the following properties: (1) For d=3 : T d (u)
AB - 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¿8 1 t (d-2)/d logP(|W a 1 (ct)nW a 2 (ct)|=t)=-I ¿ a d (c) and derive a variational representation for the rate constant I ¿ a d (c) . Here, ¿ a is the Newtonian capacity of the ball with radius a . We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to "form a Swiss cheese": the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1/d according to a certain optimal profile.
We study in detail the function c¿I ¿ a d (c) . It turns out that I ¿ a d (c)=T d (¿ a c)/¿ a , where T d has the following properties: (1) For d=3 : T d (u)
U2 - 10.4007/annals.2004.159.741
DO - 10.4007/annals.2004.159.741
M3 - Article
SN - 0003-486X
VL - 159
SP - 741
EP - 782
JO - Annals of Mathematics
JF - Annals of Mathematics
IS - 2
ER -