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.References
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Additional Information
- Yaozhong Hu
- Affiliation: Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045
- Email: yhu@ku.edu
- Khoa Lê
- Affiliation: Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045
- MR Author ID: 1036588
- Email: khoale@ku.edu
- Received by editor(s): May 5, 2014
- Received by editor(s) in revised form: March 15, 2015
- Published electronically: May 25, 2016
- Additional Notes: The first author was partially supported by a grant from the Simons Foundation #209206 and by a General Research Fund of the University of Kansas.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 1935-2002
- MSC (2010): Primary 60H30; Secondary 60H10, 60H15, 60H07, 60G17
- DOI: https://doi.org/10.1090/tran/6774
- MathSciNet review: 3581224