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**On the volume of the intersection of two Wiener sausages.**
*(English)*
Zbl 1165.60316

Summary: For \(a>0\), let \(W_1^a(t)\) and \(W_2^a(t)\) be the \(a\)-neighbourhoods of two independent standard Brownian motions in \(\mathbb R^d\) starting at 0 and observed until time \(t\). We prove that, for \(d\geq 3\) and \(c>0\),

\[ \lim_{t\to\infty} \frac{1}{(d-2)/d} \log P\big(|W_1^a(ct)\cap W_2^a(ct)|\geq t\big)= -I_d^{\kappa_a}(c) \]

and derive a variational representation for the rate constant \(I_d^{\kappa_a}(c)\). Here, \(\kappa_a\) is the Newtonian capacity of the ball with radius \(a\). We show that the optimal strategy to realise the above large deviation is for \(W_1^a(ct)\) and \(W_2^a(ct)\) 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 \(c\mapsto I_d^{\kappa_a}(c)\). It turns out that \(I_d^{\kappa_a}(c)= \Theta_d(\kappa_ac)/\kappa_a\), where \(\Theta_d\) has the following properties: (1) For \(d\geq 3\): \(\Theta_d(u)<\infty\) if and only if \(u\in(u_\diamondsuit,\infty)\), with \(u_\diamondsuit\) a universal constant; (2) For \(d=3\): \(\Theta_d\) is strictly decreasing on \((u_\diamondsuit,\infty)\) with a zero limit; (3) For \(d=4\): \(\Theta_d\) is strictly decreasing on \((u_\diamondsuit,\infty)\) with a nonzero limit; (4) For \(d\geq 5\): \(\Theta_d\) is strictly decreasing on \((u_\diamondsuit,u_d)\) and a nonzero constant on \([u_d,\infty)\), with \(u_d\) a constant depending on \(d\) that comes from a variational problem exhibiting “leakage”. This leakage is interpreted as saying that the two Wiener sausages form their intersection until time \(c^*t\), with \(c^*= u_d/\kappa_a\), and then wander off to infinity in different directions. Thus, \(c^*\) plays the role of a critical time horizon in \(d\geq 5\).

We also derive the analogous result for \(d=2\), namely,

\[ \lim_{t\to\infty} \frac{1}{\log t} \log P\big(|W_1^a(ct)\cap W_2^a(ct)|\geq t/\log t\big)= -I_2^{2\pi}(c), \]

where the rate constant has the same variational representation as in \(d\geq 3\) after \(\kappa_a\) is replaced by \(2\pi\). In this case \(I_2^{2\pi}(c)= \Theta_2(2\pi c)/2\pi\) with \(\Theta_2(u)<\infty\) if and only if \(u\in(u_\diamondsuit,\infty)\) and \(\Theta_2\) is strictly decreasing on \((u_\diamondsuit,\infty)\) with a zero limit.

\[ \lim_{t\to\infty} \frac{1}{(d-2)/d} \log P\big(|W_1^a(ct)\cap W_2^a(ct)|\geq t\big)= -I_d^{\kappa_a}(c) \]

and derive a variational representation for the rate constant \(I_d^{\kappa_a}(c)\). Here, \(\kappa_a\) is the Newtonian capacity of the ball with radius \(a\). We show that the optimal strategy to realise the above large deviation is for \(W_1^a(ct)\) and \(W_2^a(ct)\) 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 \(c\mapsto I_d^{\kappa_a}(c)\). It turns out that \(I_d^{\kappa_a}(c)= \Theta_d(\kappa_ac)/\kappa_a\), where \(\Theta_d\) has the following properties: (1) For \(d\geq 3\): \(\Theta_d(u)<\infty\) if and only if \(u\in(u_\diamondsuit,\infty)\), with \(u_\diamondsuit\) a universal constant; (2) For \(d=3\): \(\Theta_d\) is strictly decreasing on \((u_\diamondsuit,\infty)\) with a zero limit; (3) For \(d=4\): \(\Theta_d\) is strictly decreasing on \((u_\diamondsuit,\infty)\) with a nonzero limit; (4) For \(d\geq 5\): \(\Theta_d\) is strictly decreasing on \((u_\diamondsuit,u_d)\) and a nonzero constant on \([u_d,\infty)\), with \(u_d\) a constant depending on \(d\) that comes from a variational problem exhibiting “leakage”. This leakage is interpreted as saying that the two Wiener sausages form their intersection until time \(c^*t\), with \(c^*= u_d/\kappa_a\), and then wander off to infinity in different directions. Thus, \(c^*\) plays the role of a critical time horizon in \(d\geq 5\).

We also derive the analogous result for \(d=2\), namely,

\[ \lim_{t\to\infty} \frac{1}{\log t} \log P\big(|W_1^a(ct)\cap W_2^a(ct)|\geq t/\log t\big)= -I_2^{2\pi}(c), \]

where the rate constant has the same variational representation as in \(d\geq 3\) after \(\kappa_a\) is replaced by \(2\pi\). In this case \(I_2^{2\pi}(c)= \Theta_2(2\pi c)/2\pi\) with \(\Theta_2(u)<\infty\) if and only if \(u\in(u_\diamondsuit,\infty)\) and \(\Theta_2\) is strictly decreasing on \((u_\diamondsuit,\infty)\) with a zero limit.