Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extensions of Brownian motion to a family of Grushin-type singularities   We consider a one-parameter family of Grushin-type singularities on surfaces,
and discuss the possible diffusions that extend Brownian motion to the
singularity. This gives a quick proof and clear intuition for the fact that
heat can only cross the singularity for an intermediate range of the parameter.
When crossing is possible and the singularity consists of one point, we give a
complete description of these diffusions, and we describe a "best" extension,
which respects the isometry group of the surface and also realizes the unique
symmetric one-point extension of the Brownian motion, in the sense of
Chen-Fukushima. This extension, however, does not correspond to the bridging
extension, which was introduced by Boscain-Prandi, when they previously
considered self-adjoint extensions of the Laplace-Beltrami operator on the
Riemannian part for these surfaces. We clarify that several of the extensions
they considered induce diffusions that are carried by the Marin
compactification at the singularity, which is much larger than the (one-point)
metric completion. In the case when the singularity is more than one-point, a
complete classification of diffusions extending Brownian motion would be
unwieldy. Nonetheless, we again describe a "best" extension which respects the
isometry group, and in this case, this diffusion corresponds to the bridging
extension. A prominent role is played by Bessel processes (of every real
dimension) and the classical theory of one-dimensional diffusions and their
boundary conditions.

20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex