Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A PDE hierarchy for directed polymers in random environments   For a Brownian directed polymer in a Gaussian random environment, with
$q(t,\cdot)$ denoting the quenched endpoint density and \[
Q_n(t,x_1,\ldots,x_n)=\mathbf{E}[q(t,x_1)\ldots q(t,x_n)], \] we derive a
hierarchical PDE system satisfied by $\{Q_n\}_{n\geq1}$. We present two
applications of the system: (i) we compute the generator of
$\{\mu_t(dx)=q(t,x)dx\}_{t\geq0}$ for some special functionals, where
$\{\mu_t(dx)\}_{t\geq0}$ is viewed as a Markov process taking values in the
space of probability measures; (ii) in the high temperature regime with $d\geq
3$, we prove a quantitative central limit theorem for the annealed endpoint
distribution of the diffusively rescaled polymer path. We also study a nonlocal
diffusion-reaction equation motivated by the generator and establish a
super-diffusive $O(t^{2/3})$ scaling.

17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol