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Nonlinear Young integrals and differential systems in Hölder media

Authors: Yaozhong Hu and Khoa Lê
Journal: Trans. Amer. Math. Soc. 369 (2017), 1935-2002
MSC (2010): Primary 60H30; Secondary 60H10, 60H15, 60H07, 60G17
Published electronically: May 25, 2016
MathSciNet review: 3581224
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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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Additional Information

Yaozhong Hu
Affiliation: Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045

Khoa Lê
Affiliation: Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045

Keywords: Gaussian random field, sample path property, majorizing measure, nonlinear Young integral, nonlinear It\^o-Skorohod integral, transport equation, stochastic parabolic equation, multiplicative noise, Feynman-Kac formula, Malliavin calculus, diffusion process, exponential integrability of the H\"older norm of diffusion process
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.
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