The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian H\"ormander theory

TL;DR

This paper develops a covariance measure-based criterion for stochastic calculus of Gaussian processes using rough path analysis, with applications to random Fourier series, SPDEs, and non-Markovian H"ormander theory.

Contribution

It introduces a new Jain-Monrad criterion applicable without explicit covariance formulas, enabling analysis of complex Gaussian processes and their rough path properties.

Findings

Verified the criterion for many Gaussian processes including random Fourier series.

Established convergence rates for rough path approximations of Fourier series.

Applied the framework to SPDEs and non-Markovian H"ormander problems.

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 It\^o-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 H\"ormander theory.