Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Champagne subdomains with unavoidable bubbles   A champagne subdomain of a connected open set $U\ne\emptyset$ in $R^d$, $d\ge
2$, is obtained omitting pairwise disjoint closed balls $\bar{B}(x,r_x)$, $x\in
X$, the bubbles, where $X$ is an infinite, locally finite set in $U$. The union
$A$ of these balls may be unavoidable, that is, Brownian motion, starting in
$U\setminus A$ and killed when leaving $U$, may hit $A$ almost surely or,
equivalently, $A$ may have harmonic measure one for $U\setminus A$.
  Recent publications by Gardiner/Ghergu ($d\ge 3$) and by Pres ($d=2$) give
rather sharp answers to the question how small such a set $A$ may be, when $U$
is the unit ball.
  In this paper, using a totally different approach, optimal results are
obtained, results which hold as well for arbitrary connected open sets $U$.

9
The categories of this paper are: nlin
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas