Nonlinear Young integrals and differential systems in H\"older media

For H"older continuous functions $W(t,x)$ and ${\varphi }_t$, we define nonlinear integral ${\int }_a^{b}W(dt,{\varphi }_t)$ 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 $\dot \phi_t=(\partial _tW)(t, \phi_t)$isalsostudiedanditsapplicationstothetransportequation$\partial _t u(t,x)-\partial _t W(t,x)\nabla u(t,x)=0$ in rough media is given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients ${\mathrm{\partial }}_{t}u(t,x)+Lu(t,x)+u(t,x)W(t,x)=0$ 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.