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What is the probability of intersecting the set of Brownian double points? (English) Zbl 1131.60071

Let \((\Omega,\{{\mathcal F}_t\}, \mathbf{P},\{{B}_t\})\) be a planar Brownian motion starting from the point \(\rho:=(1,0)\) and let \( {\mathcal D}:=\{x: B_r=B_s=x \;\text{for some} \;0<r<s<\tau_*\} \) where \(\tau_*:=\inf\{t:|B_t|=3\). The authors introduce first the notion of two-gauge capacity and use it to prove an estimate for \(\mathbf{P}({\mathcal D}\cap A \neq \emptyset)\) for any closed subset of the disk \(\{x:|x|\leq 1/3\}\). They also prove a polar decomposition w.r.t \(\mathcal D\), for any compact subset \(A\) not containing \(\rho\). Namely, \(A\) may be written as \(A=A_1\cup A_2\) such that (1) \(A_1\) is almost surely disjoint from \(\mathcal D\), and (2) if the hitting time \(\tau_2\) of \(A_2\) is finite, then for any \(\varepsilon>0\), with probability 1, the Brownian motion stopped at time \(\tau_2+ \varepsilon\) has a double point in \(A_2\). It follows from this that \(\mathbf{P}({\mathcal D}\cap A \neq \emptyset) =\mathbf{P}(\text{Brownian motion hits}\;A_2)\).

MSC:

60J45 Probabilistic potential theory
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
60J65 Brownian motion
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References:

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