The JainMonrad criterion for rough paths and applications to random Fourier series and nonMarkovian Hörmander theory
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) 4657]. 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 CameronMartin paths and complementary Young regularity (CYR) of the CameronMartin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Itôlike probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of nonMarkovian Hörmander theory.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 arXiv:
 arXiv:1307.3460
 Bibcode:
 2013arXiv1307.3460F
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs
 EPrint:
 Published at http://dx.doi.org/10.1214/14AOP986 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)