Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On time scales and quasi-stationary distributions for multitype
  birth-and-death processes   We consider a class of birth-and-death processes describing a population made
of $d$ sub-populations of different types which interact with one another. The
state space is $\mathbb{Z}_+^d$ (unbounded). We assume that the population goes
almost surely to extinction, so that the unique stationary distribution is the
Dirac measure at the origin. These processes are parametrized by a scaling
parameter $K$ which can be thought as the order of magnitude of the total size
of the population at time $0$. For any fixed finite time span, it is well-known
that such processes, when renormalized by $K$, are close, in the limit
$K\to+\infty$, to the solutions of a certain differential equation in
$\mathbb{R}_+^d$ whose vector field is determined by the birth and death rates.
We consider the case where there is a unique attractive fixed point (off the
boundary of the positive orthant) for the vector field (while the origin is
repulsive). What is expected is that, for $K$ large, the process will stay in
the vicinity of the fixed point for a very long time before being absorbed at
the origin. To precisely describe this behavior, we prove the existence of a
quasi-stationary distribution (qsd). In fact, we establish a bound for the
total variation distance between the process conditioned to non-extinction
before time $t$ and the qsd. This bound is exponentially small in $t$, for
$t\gg \log K$. As a by-product, we obtain an estimate for the mean time to
extinction in the qsd. We also quantify how close is the law of the process
(not conditioned to non-extinction) either to the Dirac measure at the origin
or to the qsd, for times much larger than $\log K$ and much smaller than the
mean time to extinction, which is exponentially large as a function of $K$. Let
us stress that we are interested in what happens for finite $K$. We obtain
results much beyond what large deviation techniques could provide.

9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys