## 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:

 [1] Benjamini, I., Pemantle, R. and Peres, Y. (1995). Martin capacity for Markov chains. Ann. Probab. 23 1332–1346. · Zbl 0840.60068 · doi:10.1214/aop/1176988187 [2] Carleson, L. (1967). Selected Problems on Exceptional Sets . Van Nostrand, Princeton–Toronto–London. · Zbl 0189.10903 [3] Fitzsimmons, P. and Salisbury, T. (1989). Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 25 325–350. · Zbl 0689.60071 [4] Pemantle, R. and Peres, Y. (1995). Galton–Watson trees with the same mean have the same polar sets. Ann. Probab. 23 1102–1124. · Zbl 0833.60085 · doi:10.1214/aop/1176988175 [5] Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177 417–434. · Zbl 0851.60080 · doi:10.1007/BF02101900 [6] Salisbury, T. (1996). Energy, and intersections of Markov chains. In Random Discrete Structures 213–225. IMA Vol. Math. Appl. 76 . Springer, New York. · Zbl 0845.60068
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