# The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

@article{Friz2013TheJC,
title={The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian H{\"o}rmander theory},
author={Peter K. Friz and Benjamin Gess and Archil Gulisashvili and Sebastian Riedel},
journal={Annals of Probability},
year={2013},
volume={44},
pages={684-738}
}
• Published 12 July 2013
• Mathematics, Computer Science
• Annals of Probability
We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46–57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions…

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## References

SHOWING 1-10 OF 55 REFERENCES

• Mathematics
• 2010
We develop a stochastic analysis for a Gaussian process $X$ with singular covariance by an intrinsic procedure focusing on several examples such as covariance measure structure processes,
• Mathematics, Computer Science
• 2012
This paper derives explicit distance bounds for Stratonovich iterated integrals along two Gaussian processes based on the regularity assumption of their covariance functions, and shows how to recover the optimal time regularity for solutions of some rough SPDEs.
The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to
• Mathematics
• 2010
We prove a generalisation of Fernique's theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough
This paper aims to provide a systematic approach to the treatment of differential equations of the type dyt = Si fi(yt) dxti where the driving signal xt is a rough path. Such equations are very
• Mathematics
• 2011
Under the key assumption of finite {\rho}-variation, {\rho}\in[1,2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are
• Computer Science
SIAM J. Numer. Anal.
• 2016
This paper investigates, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense, and focuses on numerical implementations, and more specifically on the savings possible via multilevel methods.
• Mathematics
• 2015
We consider stochastic differential equations driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields satisfy Hormander's bracket condition, we demonstrate that
• Mathematics
• 2014
We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on