Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stochastic particle approximation of the Keller-Segel equation and
  two-dimensional generalization of Bessel processes   The Keller-Segel partial differential equation is a two-dimensional model for
chemotaxis. When the total mass of the initial density is one, it is known to
exhibit blow-up in finite time as soon as the sensitivity $\chi$ of bacteria to
the chemo-attractant is larger than $8\pi$. We investigate its approximation by
a system of $N$ two-dimensional Brownian particles interacting through a
singular attractive kernel in the drift term. In the very subcritical case
$\chi\textless{}2\pi$, the diffusion strongly dominates this singular drift: we
obtain existence for the particle system and prove that its flow of empirical
measures converges, as $N\to\infty$ and up to extraction of a subsequence, to a
weak solution of the Keller-Segel equation. We also show that for any $N\ge 2$
and any value of $\chi\textgreater{}0$, pairs of particles do collide with
positive probability: the singularity of the drift is indeed visited.
Nevertheless, when $\chi\textless{}2\pi N$, it is possible to control the drift
and obtain existence of the particle system until the first time when at least
three particles collide. We check that this time is a.s. infinite, so that
global existence holds for the particle system, if and only if $\chi\leq
8\pi(N-2)/(N-1)$. Finally, we remark that in the system with $N=2$ particles,
the difference between the two positions provides a natural two-dimensional
generalization of Bessel processes, which we study in details.

37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
16
The categories of this paper are: stat