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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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

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

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

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Zentralblatt MATH identifier

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.)

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


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.

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