Nonlinear Young integrals and differential systems in Hölder media

Hu, Yaozhong; Lê, Khoa

doi:10.1090/tran/6774

Nonlinear Young integrals and differential systems in Hölder media
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by Yaozhong Hu and Khoa Lê PDF

Trans. Amer. Math. Soc. 369 (2017), 1935-2002 Request permission

Abstract:

For Hölder continuous random field $W(t,x)$ and stochastic process $\varphi _t$, we define nonlinear integral $\int _a^b W(dt, \varphi _t)$ 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 $\dot \varphi _t=(\partial _tW)(t, \varphi _t)$ 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 are given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients $\partial _t u(t,x)+Lu(t,x) +u(t,x) \partial _t W(t,x)=0$ 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 multi- dimensional rectangles by majorizing measures.

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