Journal
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 369, Issue 3, Pages 1935-2002Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/tran/6774
Keywords
Gaussian random field; sample path property; majorizing measure; nonlinear Young integral; nonlinear Ito-Skorohod integral; transport equation; stochastic parabolic equation; multiplicative noise; Feynman-Kac formula; Malliavin calculus; diffusion process; exponential integrability of the Holder norm of diffusion process
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Funding
- Simons Foundation [209206]
- General Research Fund of the University of Kansas
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For Holder continuous random field W(t, x) and stochastic process phi(t), we define nonlinear integral integral(b)(a) W(dt, phi(t)) in various senses, including pathwise and Ito-Skorohod. We study their properties and relations. The stochastic flow in a time dependent rough vector field associated with (phi) over dot(t) = (partial derivative W-t)(t, phi(t)) is also studied, and its applications to the transport equation partial derivative(t)u(t, x) - partial derivative W-t(t, x)del u(t, x) = 0 in rough media are given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients partial derivative(t)u(t, x) + Lu(t, x) + u(t, x)partial derivative W-t(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 multidimensional rectangles by majorizing measures.
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