Hörmander’s theorem for semilinear SPDEs

We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hormander's bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris's lemma to work directly on the Malliavin matrix, instead of the "reduced Malliavin matrix" which is not available in this context. On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild Ito formula for rough PDEs.

[1]  C. S. G. David.  Stochastic Analysis , 2021, Nature.

[2]  Jonathan C. Mattingly,et al.  The strong Feller property for singular stochastic PDEs , 2016, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[3]  Paul Gassiat,et al.  Malliavin Calculus for regularity structures: the case of gPAM , 2015, 1511.08888.

[4]  Terry Lyons,et al.  The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations , 2015, 1503.06175.

[5]  M. Gubinelli,et al.  Unbounded rough drivers , 2015, 1501.02074.

[6]  W. Stannat,et al.  Stochastic partial differential equations: a rough path view , 2014, 1412.6557.

[7]  Martin Hairer.  Introduction to regularity structures , 2014, Universitext.

[8]  Martin Hairer,et al.  Geometric versus non-geometric rough paths , 2012, 1210.6294.

[9]  Nicolas Perkowski,et al.  PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES , 2012, Forum of Mathematics, Pi.

[10]  Martin Hairer,et al.  Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths , 2011, 1104.5218.

[11]  S. Riedel,et al.  Integrability of (Non-)Linear Rough Differential Equations and Integrals , 2011, 1104.0577.

[12]  Martin Hairer,et al.  On Malliavinʼs proof of Hörmanderʼs theorem , 2011, 1103.1998.

[13]  S. Tindel,et al.  Malliavin calculus for fractional heat equation , 2009, 1109.0422.

[14]  M. Gubinelli,et al.  Non-linear rough heat equations , 2009, 0911.0618.

[15]  N. Pillai,et al.  Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion , 2009, 0909.4505.

[16]  Martin Hairer,et al.  An Introduction to Stochastic PDEs , 2009, 0907.4178.

[17]  Martin Hairer,et al.  A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs , 2008, 0808.1361.

[18]  Massimiliano Gubinelli,et al.  Rough evolution equations , 2008, 0803.0552.

[19]  T. Kurtz,et al.  Stochastic equations in infinite dimensions , 2006 .

[20]  Jonathan C. Mattingly,et al.  Malliavin calculus for infinite-dimensional systems with additive noise , 2006, math/0610754.

[21]  Fabrice Baudoin,et al.  Hypoellipticity in infinite dimensions and an application in interest rate theory , 2005, math/0508452.

[22]  Jonathan C. Mattingly,et al.  Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing , 2004, math/0406087.

[23]  M. Gubinelli.  Controlling rough paths , 2003, math/0306433.

[24]  Terry Lyons,et al.  System Control and Rough Paths , 2003 .

[25]  J. Eckmann,et al.  Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise , 2000, nlin/0009028.

[26]  Sandra Cfrrai.  Ergodicity for stochastic reaction-diffusion systems with polynomial coefficients , 1999 .

[27]  Terry Lyons.  Di erential equations driven by rough signals , 1998 .

[28]  K. Elworthy.  ERGODICITY FOR INFINITE DIMENSIONAL SYSTEMS (London Mathematical Society Lecture Note Series 229) By G. Da Prato and J. Zabczyk: 339 pp., £29.95, LMS Members' price £22.47, ISBN 0 521 57900 7 (Cambridge University Press, 1996). , 1997 .

[29]  Franco Flandoli,et al.  Ergodicity of the 2-D Navier-Stokes equation under random perturbations , 1995 .

[30]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[31]  R. Temam,et al.  Inertial Forms of Navier-Stokes Equations on the Sphere , 1993, chao-dyn/9304004.

[32]  D. Stroock,et al.  Applications of the Malliavin calculus. II , 1985 .

[33]  J. Bismut.  Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions , 1981 .

[34]  L. C. Young,et al.  An inequality of the Hölder type, connected with Stieltjes integration , 1936 .

[35]  Martin Hairer,et al.  A Course on Rough Paths , 2020, Universitext.

[36]  David Nualart Rodón,et al.  The Malliavin Calculus and Related Topics , 2006 .

[37]  J. Zabczyk,et al.  Strong feller property for stochastic semilinear equations , 1995 .

[38]  J. Norris.  Simplified Malliavin calculus , 1986 .

[39]  P. Malliavin.  Stochastic calculus of variation and hypoelliptic operators , 1978 .

[40]  Kuo-Tsai Chen,et al.  Iterated Integrals and Exponential Homomorphisms , 1954 .