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The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory. (English) Zbl 1347.60097

The authors study stochastic calculus for an extended class of Gaussian processes, based on rough path analysis. Starting with the properties of variations, they introduce the property of “complementary Young regularity” (CYR) which is known to imply Malliavin regularity and also Itô-like probabilistic estimates for stochastic integrals despite their rough pathwise construction. The connection of CYR to Cameron-Martin regularity is studied. It is established that given a multidimensional Gaussian process with covariance of finite \(\rho\)-variation, \(\rho<2\), then CYR holds. The key condition for CYR is a covariance measure structure combined with a classical Jain-Monrad criterion for rough paths. This key condition is checked for many processes, including random Fourier series, where the covariance itself is not known explicitly, but only given as a Fourier series. The rate of convergence of natural approximations of rough paths given in terms of Fourier multipliers is estimated. An application to stochastic partial differential equations, including fractional stochastic heat equations, is given. An application in the context of non-Markovian Hörmander theory is also discussed.

MSC:

60H99 Stochastic analysis
60G15 Gaussian processes
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G17 Sample path properties
42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
42A45 Multipliers in one variable harmonic analysis
42A61 Probabilistic methods for one variable harmonic analysis
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[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. · Zbl 1123.60038 · doi:10.1007/s00440-006-0035-0
[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 · Zbl 1343.60097 · doi:10.1137/140995209
[3] Cass, T. and Friz, P. (2010). Densities for rough differential equations under Hörmander’s condition. Ann. of Math. (2) 171 2115-2141. · Zbl 1205.60105 · doi:10.4007/annals.2010.171.2115
[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. · Zbl 1175.60034 · doi:10.1090/S0002-9947-09-04677-7
[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. · Zbl 1309.60055 · doi:10.1214/13-AOP896
[6] Cass, T., Litterer, C. and Lyons, T. (2013). Integrability and tail estimates for Gaussian rough differential equations. Ann. Probab. 41 3026-3050. · Zbl 1278.60091 · doi:10.1214/12-AOP821
[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. · Zbl 1296.60097 · doi:10.1214/12-AAP896
[8] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44 . Cambridge Univ. Press, Cambridge. · Zbl 0761.60052
[9] Decreusefond, L. (2005). Stochastic integration with respect to Volterra processes. Ann. Inst. Henri Poincaré Probab. Stat. 41 123-149. · Zbl 1071.60040 · doi:10.1016/j.anihpb.2004.03.004
[10] Diehl, J., Friz, P. and Stannat, W. (2014). Stochastic partial differential equations: A rough path view. Preprint. Available at . arXiv:1412.6557 · Zbl 1392.35335
[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. · Zbl 1304.60065 · doi:10.1016/j.spa.2014.08.005
[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. · Zbl 0937.28001
[13] Friz, P. and Oberhauser, H. (2010). A generalized Fernique theorem and applications. Proc. Amer. Math. Soc. 138 3679-3688. · Zbl 1202.60057 · doi:10.1090/S0002-9939-2010-10528-2
[14] Friz, P. and Riedel, S. (2013). Integrability of (non-)linear rough differential equations and integrals. Stoch. Anal. Appl. 31 336-358. · Zbl 1274.60173 · doi:10.1080/07362994.2013.759758
[15] Friz, P. and Riedel, S. (2014). Convergence rates for the full Gaussian rough paths. Ann. Inst. Henri Poincaré Probab. Stat. 50 154-194. · Zbl 1295.60045 · doi:10.1214/12-AIHP507
[16] Friz, P. and Victoir, N. (2006). A variation embedding theorem and applications. J. Funct. Anal. 239 631-637. · Zbl 1114.46022 · doi:10.1016/j.jfa.2005.12.021
[17] Friz, P. and Victoir, N. (2010). Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 369-413. · Zbl 1202.60058 · doi:10.1214/09-AIHP202
[18] Friz, P. and Victoir, N. (2011). A note on higher dimensional \(p\)-variation. Electron. J. Probab. 16 1880-1899. · Zbl 1244.60066 · doi:10.1214/EJP.v16-951
[19] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths with an Introduction to Regularity Structures . Springer, Berlin. · Zbl 1327.60013 · doi:10.1007/978-3-319-08332-2
[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. · Zbl 1193.60053 · doi:10.1017/CBO9780511845079
[21] Gubinelli, M., Imkeller, P. and Perkowski, N. (2012). Paraproducts, rough paths and controlled distributions. Available at . arXiv:1210.2684 · Zbl 1349.00120 · doi:10.4171/OWR/2012/41
[22] Hairer, M. (2011). Rough stochastic PDEs. Comm. Pure Appl. Math. 64 1547-1585. · Zbl 1229.60079 · doi:10.1002/cpa.20383
[23] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559-664. · Zbl 1281.60060 · doi:10.4007/annals.2013.178.2.4
[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. · Zbl 1221.60083 · doi:10.1214/10-AIHP377
[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. · Zbl 1140.60329 · doi:10.1214/07-AAP441
[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. · Zbl 1138.60326 · doi:10.4310/CMS.2005.v3.n4.a8
[27] Hairer, M. and Weber, H. (2013). Rough Burgers-like equations with multiplicative noise. Probab. Theory Related Fields 155 71-126. · Zbl 1303.60055 · doi:10.1007/s00440-011-0392-1
[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. · Zbl 0521.35001
[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. · Zbl 1046.60033 · doi:10.1090/conm/336/06034
[30] Jain, N. C. and Monrad, D. (1983). Gaussian measures in \(B_{p}\). Ann. Probab. 11 46-57. · Zbl 0504.60045 · doi:10.1214/aop/1176993659
[31] Kahane, J.-P. (1985). Some Random Series of Functions , 2nd ed. Cambridge Studies in Advanced Mathematics 5 . Cambridge Univ. Press, Cambridge. · Zbl 0571.60002
[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. · Zbl 0677.42001
[34] Krasniqi, X. Z. (2011). On the second derivative of the sums of trigonometric series. An. Univ. Craiova Ser. Mat. Inform. 38 76-86. · Zbl 1265.42013
[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. · Zbl 1126.60046 · doi:10.1016/j.jfa.2007.03.031
[37] Lorentz, G. G. (1948). Fourier-Koeffizienten und Funktionenklassen. Math. Z. 51 135-149. · Zbl 0030.24903 · doi:10.1007/BF01290998
[38] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths . Oxford Univ. Press, Oxford. Oxford Science Publications. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001
[39] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215-310. · Zbl 0923.34056 · doi:10.4171/RMI/240
[40] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908 . Springer, Berlin. · Zbl 1176.60002 · doi:10.1007/978-3-540-71285-5
[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. · Zbl 1129.60002 · doi:10.1017/CBO9780511617997
[42] Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Probability and Its Applications ( New York ). Springer, Berlin. · Zbl 1099.60003 · doi:10.1007/3-540-28329-3
[43] Riedel, S. and Xu, W. (2013). A simple proof of distance bounds for Gaussian rough paths. Electron. J. Probab. 18 no. 108, 22. · Zbl 1290.60046 · doi:10.1214/EJP.v18-2387
[44] Russo, F. and Tudor, C. A. (2006). On bifractional Brownian motion. Stochastic Process. Appl. 116 830-856. · Zbl 1100.60019 · doi:10.1016/j.spa.2005.11.013
[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). · Zbl 0997.26007
[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. · Zbl 0608.60060
[48] Zygmund, A. (1959). Trigonometric Series , 2nd ed. Vols. I , II . Cambridge Univ. Press, New York. · Zbl 0085.05601
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