NONLINEAR YOUNG INTEGRALS AND DIFFERENTIAL SYSTEMS IN HOLDER MEDIA

For Holder continuous random field W(t, x) and stochastic process phi(t), we define nonlinear integral integral(b)(a) W(dt, phi(t)) in various senses, including pathwise and Ito-Skorohod. We study their properties and relations. The stochastic flow in a time dependent rough vector field associated with (phi) over dot(t) = (partial derivative W-t)(t, phi(t)) is also studied, and its applications to the transport equation partial derivative(t)u(t, x) - partial derivative W-t(t, x)del u(t, x) = 0 in rough media are given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients partial derivative(t)u(t, x) + Lu(t, x) + u(t, x)partial derivative W-t(t, x) = 0 is given, where L is a second order elliptic differential operator with random coefficients (dependent on 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 and on the coefficients of L. Along the way, we also obtain an upper bound for increments of stochastic processes on multidimensional rectangles by majorizing measures.