The Annals of Probability

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

Peter K. Friz, Benjamin Gess, Archil Gulisashvili, and Sebastian Riedel

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Abstract

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 directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron–Martin paths and complementary Young regularity (CYR) of the Cameron–Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Itô-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hörmander theory.

Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 684-738.

Dates
Received: November 2013
Revised: July 2014
First available in Project Euclid: 2 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1454423053

Digital Object Identifier
doi:10.1214/14-AOP986

Mathematical Reviews number (MathSciNet)
MR3456349

Zentralblatt MATH identifier
1347.60097

Subjects
Primary: 60G15: Gaussian processes 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G17: Sample path properties 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)

Keywords
Gaussian processes rough paths Cameron–Martin regularity random Fourier series fractional stochastic heat equation SPDE

Citation

Friz, Peter K.; Gess, Benjamin; Gulisashvili, Archil; Riedel, Sebastian. The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory. Ann. Probab. 44 (2016), no. 1, 684--738. doi:10.1214/14-AOP986. https://projecteuclid.org/euclid.aop/1454423053


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