Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Brownian motion between two random trajectories   Consider the first exit time of one-dimensional Brownian motion
$\{B_s\}_{s\geq 0}$ from a random passageway. We discuss a Brownian motion with
two time-dependent random boundaries in quenched sense. Let $\{W_s\}_{s\geq 0}$
be an other one-dimensional Brownian motion independent of $\{B_s\}_{s\geq 0}$
and let $\bfP(\cdot|W)$ represent the conditional probability depending on the
realization of $\{W_s\}_{s\geq 0}$. We show that
$$-t^{-1}\ln\bfP^x(\forall_{s\in[0,t]}a+\beta W_s\leq B_s\leq b+\beta W_s|W)$$
converges to a finite positive constant $\gamma(\beta)(b-a)^{-2}$ almost surely
and in $L^p~ (p\geq 1)$ if $a<B_0=x<b$ and $W_0=0.$ When $\beta=1, a+b=2x,$ it
is equivalent to the random small ball probability problem in the sense of
equiditribution, which has been investigated in \cite{DL2005}. We also find
some properties of the function $\gamma(\beta)$. An important moment estimation
has also been obtained, which can be applied to discuss the small deviation of
random walk with random environment in time (see [12]).

11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol