  For H\"older continuous functions $W(t,x)$ and $\phi_t$, we define nonlinear
integral $\int_a^b W(dt, \phi_t)$ in various senses, including It\^o-Skorohod
and pathwise. We study their properties and relations. The stochastic flow in a
time dependent rough vector field associated with $\dot \phi_t=(\partial
_tW)(t, \phi_t)$ is also studied and its applications to the transport equation
$\partial _t u(t,x)-\partial _t W(t,x)\nabla u(t,x)=0$ in rough media is given.
The Feynman-Kac solution to the stochastic partial differential equation with
random coefficients $\partial _t u(t,x)+Lu(t,x) +u(t,x)W(t,x)=0$ are given,
where $L$ is a second order elliptic differential operator with random
coefficients (dependent on $W$). To establish such 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 $W$. Along the way,
we also obtain an upper bound for increments of stochastic processes on
multidimensional rectangles by majorizing measures.

Probability;Analysis of PDEs