Nonlinear Young integrals and differential systems in Hölder media

Source abstract

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.