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

TL;DR

This paper develops nonlinear Young integrals for H"older continuous functions, studies associated stochastic flows, and applies these to solve rough media PDEs and SPDEs, providing new formulas and integrability results.

Contribution

It introduces new definitions of nonlinear integrals in rough media, analyzes their properties, and applies them to stochastic flows and PDEs with random coefficients, advancing the theory of rough stochastic systems.

Findings

Defined nonlinear Young integrals in various senses.

Established properties and relations of these integrals.

Derived exponential integrability under mild covariance conditions.

Abstract

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.