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

Full-text: Open access


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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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.

Export citation


  • [1] Baudoin, F. and Hairer, M. (2007). A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 373–395.
  • [2] Bayer, C., Friz, P. K., Riedel, S. and Schoenmakers, J. (2013). From rough paths estimates to multilevel Monte Carlo. Available at arXiv:1305.5779.
  • [3] Cass, T. and Friz, P. (2010). Densities for rough differential equations under Hörmander’s condition. Ann. of Math. (2) 171 2115–2141.
  • [4] Cass, T., Friz, P. and Victoir, N. (2009). Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 3359–3371.
  • [5] Cass, T., Hairer, M., Litterer, C. and Tindel, S. (2015). Smoothness of the density for solutions to Gaussian rough differential equations. Ann. Probab. 43 188–239.
  • [6] Cass, T., Litterer, C. and Lyons, T. (2013). Integrability and tail estimates for Gaussian rough differential equations. Ann. Probab. 41 3026–3050.
  • [7] Crisan, D., Diehl, J., Friz, P. K. and Oberhauser, H. (2013). Robust filtering: Correlated noise and multidimensional observation. Ann. Appl. Probab. 23 2139–2160.
  • [8] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [9] Decreusefond, L. (2005). Stochastic integration with respect to Volterra processes. Ann. Inst. Henri Poincaré Probab. Stat. 41 123–149.
  • [10] Diehl, J., Friz, P. and Stannat, W. (2014). Stochastic partial differential equations: A rough path view. Preprint. Available at arXiv:1412.6557.
  • [11] Diehl, J., Oberhauser, H. and Riedel, S. (2013). A Levy-area between Brownian motion and rough paths with applications to robust non-linear filtering and RPDEs. Stochastic Process. Appl. 125 161–181.
  • [12] Dudley, R. M. and Norvaiša, R. (1998). An Introduction to $p$-Variation and Young Integrals: With Emphasis on Sample Functions of Stochastic Processes. Lecture Notes/Centre for Mathematical Physics and Stochastics 1. Aarhus, Denmark.
  • [13] Friz, P. and Oberhauser, H. (2010). A generalized Fernique theorem and applications. Proc. Amer. Math. Soc. 138 3679–3688.
  • [14] Friz, P. and Riedel, S. (2013). Integrability of (non-)linear rough differential equations and integrals. Stoch. Anal. Appl. 31 336–358.
  • [15] Friz, P. and Riedel, S. (2014). Convergence rates for the full Gaussian rough paths. Ann. Inst. Henri Poincaré Probab. Stat. 50 154–194.
  • [16] Friz, P. and Victoir, N. (2006). A variation embedding theorem and applications. J. Funct. Anal. 239 631–637.
  • [17] Friz, P. and Victoir, N. (2010). Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 369–413.
  • [18] Friz, P. and Victoir, N. (2011). A note on higher dimensional $p$-variation. Electron. J. Probab. 16 1880–1899.
  • [19] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths with an Introduction to Regularity Structures. Springer, Berlin.
  • [20] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths. Cambridge Studies in Advanced Mathematics. Theory and Applications. 120. Cambridge Univ. Press, Cambridge.
  • [21] Gubinelli, M., Imkeller, P. and Perkowski, N. (2012). Paraproducts, rough paths and controlled distributions. Available at arXiv:1210.2684.
  • [22] Hairer, M. (2011). Rough stochastic PDEs. Comm. Pure Appl. Math. 64 1547–1585.
  • [23] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
  • [24] Hairer, M. and Pillai, N. S. (2011). Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 47 601–628.
  • [25] Hairer, M., Stuart, A. M. and Voss, J. (2007). Analysis of SPDEs arising in path sampling. II. The nonlinear case. Ann. Appl. Probab. 17 1657–1706.
  • [26] Hairer, M., Stuart, A. M., Voss, J. and Wiberg, P. (2005). Analysis of SPDEs arising in path sampling. I. The Gaussian case. Commun. Math. Sci. 3 587–603.
  • [27] Hairer, M. and Weber, H. (2013). Rough Burgers-like equations with multiplicative noise. Probab. Theory Related Fields 155 71–126.
  • [28] Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 256. Springer, Berlin.
  • [29] Houdré, C. and Villa, J. (2003). An example of infinite dimensional quasi-helix. In Stochastic Models (Mexico City, 2002). Contemp. Math. 336 195–201. Amer. Math. Soc., Providence, RI.
  • [30] Jain, N. C. and Monrad, D. (1983). Gaussian measures in $B_{p}$. Ann. Probab. 11 46–57.
  • [31] Kahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Studies in Advanced Mathematics 5. Cambridge Univ. Press, Cambridge.
  • [32] Kolmogorow, A. N. (1923). Sur l’ordre de grandeur des coefficient de la série de Fourier-Lebesque. Bull. Acad. Polon., Ser. A 83–86.
  • [33] Körner, T. W. (1989). Fourier Analysis, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [34] Krasniqi, X. Z. (2011). On the second derivative of the sums of trigonometric series. An. Univ. Craiova Ser. Mat. Inform. 38 76–86.
  • [35] Kruk, I. and Russo, F. (2010). Malliavin–Skorohod calculus and Paley–Wiener integral for covariance singular processes. Available at arXiv:1011.6478.
  • [36] Kruk, I., Russo, F. and Tudor, C. A. (2007). Wiener integrals, Malliavin calculus and covariance measure structure. J. Funct. Anal. 249 92–142.
  • [37] Lorentz, G. G. (1948). Fourier-Koeffizienten und Funktionenklassen. Math. Z. 51 135–149.
  • [38] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Univ. Press, Oxford. Oxford Science Publications.
  • [39] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [40] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908. Springer, Berlin.
  • [41] Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, Cambridge.
  • [42] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Springer, Berlin.
  • [43] Riedel, S. and Xu, W. (2013). A simple proof of distance bounds for Gaussian rough paths. Electron. J. Probab. 18 no. 108, 22.
  • [44] Russo, F. and Tudor, C. A. (2006). On bifractional Brownian motion. Stochastic Process. Appl. 116 830–856.
  • [45] Teljakovskiĭ, S. A. (1973). A certain sufficient condition of Sidon for the integrability of trigonometric series. Mat. Zametki 14 317–328.
  • [46] Towghi, N. (2002). Multidimensional extension of L. C. Young’s inequality. JIPAM. J. Inequal. Pure Appl. Math. 3 Article 22, 13 pp. (electronic).
  • [47] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École d’Été de Probabilités de Saint-Flour XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.
  • [48] Zygmund, A. (1959). Trigonometric Series, 2nd ed. Vols. I, II. Cambridge Univ. Press, New York.