For Hölder continuous random field
W
(
t
,
x
)
W(t,x)
and stochastic process
φ
t
\varphi _t
, we define nonlinear integral
∫
a
b
W
(
d
t
,
φ
t
)
\int _a^b W(dt, \varphi _t)
in various senses, including pathwise and Itô-Skorohod. We study their properties and relations. The stochastic flow in a time dependent rough vector field associated with
φ
˙
t
=
(
∂
t
W
)
(
t
,
φ
t
)
\dot \varphi _t=(\partial _tW)(t, \varphi _t)
is also studied, and its applications to the transport equation
∂
t
u
(
t
,
x
)
−
∂
t
W
(
t
,
x
)
∇
u
(
t
,
x
)
=
0
\partial _t u(t,x)-\partial _t W(t,x)\nabla u(t,x)=0
in rough media are given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients
∂
t
u
(
t
,
x
)
+
L
u
(
t
,
x
)
+
u
(
t
,
x
)
∂
t
W
(
t
,
x
)
=
0
\partial _t u(t,x)+Lu(t,x) +u(t,x) \partial _t W(t,x)=0
is given, where
L
L
is a second order elliptic differential operator with random coefficients (dependent on
W
W
). To establish such a formula the main difficulty is the exponential integrability of some nonlinear integrals, which is proved to be true under some mild conditions on the covariance of
W
W
and on the coefficients of
L
L
. Along the way, we also obtain an upper bound for increments of stochastic processes on multi- dimensional rectangles by majorizing measures.