Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fixed points with finite mean of the smoothing transform in random
  environments   At each time $n\in\mathbb{N}$, let
$\bar{Y}^{(n)}=(y_{1}^{(n)},y_{2}^{(n)},\cdots)$ be a random sequence of
non-negative numbers that are ultimately zero in a random environment
$\xi=(\xi_{n})_{n\in\mathbb{N}}$ in time, which satisfies for each
$n\in\mathbb{N}$ and a.e.
$\xi,~E_{\xi}[\sum_{i\in\mathbb{N}_{+}}y_{i}^{(n)}(\xi)]=1.$ The existence and
uniqueness of the non-negative fixed points of the associated smoothing
transform in random environments is considered. These fixed points are
solutions of the distributional equation for
$a.e.~\xi,~Z(\xi)\overset{d}{=}\sum_{i\in\mathbb{N}_{+}}y_{i}^{(0)}(\xi)Z_{i}(T\xi),$
where when given the environment $\xi$, $Z_{i}(T\xi)~(i\in\mathbb{N}_{+})$ are
$i.i.d.$ non-negative random variables, and distributed the same as $Z(\xi)$.
As an application, the martingale convergence of the branching random walk in
random environments is given as well. The classical results by Biggins (1977)
has been extended to the random environment situation.

13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph