Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Almost sure diffusion approximation in averaging via rough paths theory   The paper deals with the fast-slow motions setups in the continuous time
$\frac {dX^\ve(t)}{dt}=\frac 1\ve\sig(X^\ve(t))\xi(t/\ve^2)+b(X^\ve(t)),\, t\in
[0,T]$ and the discrete time
$X_N((n+1)/N)=X_N(n/N)+N^{-1/2}\sig(X_N(n/N))\xi(n))+N^{-1}b(X_N(n/N))\xi(n)$,
$n=0,1,...,[TN]$ where $\sig$ and $b$ are smooth matrix and vector functions,
respectively, $\xi$ is a centered stationary vector stochastic process and
$\ve, 1/N$ are small parameters. We derive, first, estimates in the strong
invariance principles for sums $S_{N}(t)=N^{-1/2}\sum_{0\leq k< [Nt]}\xi(k)$
and iterated sums $\bbS^{ij}_{N}(t)=N^{-1}\sum_{0\leq
k<l<[Nt]}\xi_i(k)\xi_j(l)$ together with the corresponding results for
integrals in the continuous time case which, in fact, yields almost sure
invariance principles for iterated sums and integrals of any order and,
moreover, implies laws of iterated logarithm for these objects. Then, relying
on the rough paths theory, we obtain strong almost sure approximations of
processes $X^\ve$ and $X_N$ by corresponding diffusion processes $\Xi^\ve$ and
$\Xi_N$, respectively. Previous results for the above setup dealt either with
weak or moment diffusion approximations and not with almost sure approximation
which is the new and natural generalization of well known works on strong
invariance principles for sums of weakly dependent random variables.

16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
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
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs