Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dissipation and high disorder   Given a field $\{B(x)\}_{x\in\mathbf{Z}^d}$ of independent standard Brownian
motions, indexed by $\mathbf{Z}^d$, the generator of a suitable Markov process
on $\mathbf{Z}^d,\,\,\mathcal{G},$ and sufficiently nice function
$\sigma:[0,\infty)\to[0,\infty),$ we consider the influence of the parameter
$\lambda$ on the behavior of the system, \begin{align*}
  \rm{d} u_t(x) = & (\mathcal{G}u_t)(x)\,\rm{d} t + \lambda\sigma(u_t(x))\rm{d}
B_t(x)
  \qquad[t>0,\ x\in\mathbf{Z}^d],
  &u_0(x)=c_0\delta_0(x). \end{align*}
  We show that for any $\lambda>0$ in dimensions one and two the total mass
$\sum_{x\in\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for dimensions
greater than two there is a phase transition point $\lambda_c\in(0,\infty)$
such that for $\lambda>\lambda_c,\, \sum_{\mathbf{Z}^d}u_t(x)\to 0$ as
$t\to\infty$ while for $\lambda<\lambda_c,\,\sum_{\mathbf{Z}^d}u_t(x)\not\to 0$
as $t\to\infty.$

18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph