Revista Matemática Iberoamericana

A signed measure on rough paths associated to a PDE of high order: results and conjectures

Daniel Levin and Terry Lyons

Full-text: Open access


Following old ideas of V. Yu. Krylov we consider the possibility that high order differential operators of dissipative type and constant coefficients might be associated, at least formally, with signed measures on path space in the same way that Wiener measure is associated with the Laplacian. There are fundamental difficulties with this idea because the measure would always have locally infinite mass. However, this paper provides evidence that if one considers equivalence classes of paths corresponding to distinct parameterisations of the same path, the measures might really exist on this quotient space. Precisely, we consider the measures on piecewise linear paths with given time partition defined using the semigroup associated to the differential operator and prove that these measures converge in distribution when the test functions on path space are the iterated integrals of the paths. Given a "random" piecewise-linear path, we evaluate its "expected" signature in terms of an explicit tensor series in the tensor algebra. Our approach uses an integration by parts argument under very mild conditions on the polynomial corresponding to the PDE of high order.

Article information

Rev. Mat. Iberoamericana, Volume 25, Number 3 (2009), 971-994.

First available in Project Euclid: 3 November 2009

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 60H15: Stochastic partial differential equations [See also 35R60]

high-order PDE brownian motion rough path signature of a path


Levin, Daniel; Lyons, Terry. A signed measure on rough paths associated to a PDE of high order: results and conjectures. Rev. Mat. Iberoamericana 25 (2009), no. 3, 971--994.

Export citation


  • Beghin, L., Hochberg, K. J. and Orsingher, E.: Conditional maximal distributions of processes related to higher-order heat-type equations. Stochastic Process. Appl. 85 (2000), no. 2, 209-223.
  • Cass, T. and Friz, P.: Densities for rough differential equations under Hörmander's condition. To appear in Ann. of Math.
  • Chen, K. T.: Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. (2) 65 (1957), 163-178.
  • Davies, E. B.: Spectral theory and differential operators. Cambridge Studies in Advanced Mathematics 42. Cambridge Univ. Press, Cambridge, 1995.
  • Davies, E. B.: Long time asymptotics of fourth order parabolic equations. J. Anal. Math. 67 (1995), 323-345.
  • Doob, J. L.: Semimartingales and subharmonic functions. Trans. Amer. Math. Soc. 77 (1954), 86-121.
  • Fawcett, T.: Problems in stochastic analysis. Connections between rough paths and non-commutative harmonic analysis. Ph.D. Thesis. Oxford University, 2003.
  • Hambly, B. M. and Lyons, T. J.: Uniqueness for the signature of a path of bounded variation and the reduced path group. To appear in Ann. of Math.
  • Hochberg, K. J.: A signed measure on path space related to Wiener measure. Ann. Probab. 6 (1978), no. 3, 433-458.
  • Hochberg, K. J.: Central limit theorem for signed distributions. Proc. Amer. Math. Soc. 79 (1980), no. 2, 298-302.
  • Hochberg, K. J. and Orsingher, E.: Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theoret. Probab. 9 (1996), no. 2, 511-532.
  • Hörmander, L.: The analysis of linear partial differential operators. II. Differential operators with constant coefficients. Grundlehren der Mathematischen Wissenschaften 257. Springer-Verlag, Berlin, 1983.
  • Itô, K.: Stochastic integral. Proc. Imp. Acad. Tokyo 20 (1944), 519-524.
  • Krylov, V. J.: Some properties of the distribution corresponding to the equation $\partial u/\partial t=(-1)\spq+1\partial \sp2qu/\partial x\sp2q$. Dokl. Akad. Nauk SSSR 132 1254-1257; translated as Soviet Math. Dokl. 1 (1960), 760-763.
  • Kusuoka, S. and Stroock, D.: Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391-442.
  • Kusuoka, S. and Stroock, D.: Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 1-76.
  • Kusuoka, S. and Stroock, D.: Applications of the Malliavin calculus. I. In Stochastic analysis (Katata/Kyoto, 1982), 271-306. North-Holland Math. Library 32. North-Holland, Amsterdam, 1984.
  • Kusuoka, S.: Malliavin calculus revisited. J. Math. Sci. Univ. Tokyo 10 (2003), no. 2, 261-277.
  • Levin, D. and Wildon, M.: A combinatorial method for calculating the moments of Lévy area. Trans. Amer. Math. Soc. 360 (2008), no. 12, 6695-6709.
  • Lévy, P.: Wiener's random function, and other Laplacian random functions. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, 171-187. Univ. California Press, Berkeley and Los Angeles, 1951.
  • Lojasiewicz, S.: Ensembles semi-analytiques. Cours Faculté des Sciences d'Orsay, Mimeographié I.H.E.S. Bures-sur-Yvette, July, 1965.
  • Lyons, T. and Victoir, N.: Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2041, 169-198.
  • Lyons, T. J.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215-310.
  • Lyons, T. J., Caruana, M. and Lévy, T.: Differential equations driven by rough paths. Lecture Notes in Mathematics 1908. Springer, Berlin, 2007.
  • Malliavin, P.: Infinite-dimensional analysis. Bull. Sci. Math. 117 (1993), no. 1, 63-90.
  • Mascarello, M. and Rodino, L.: Partial differential equations with multiple characteristics. Math. Topics 13. Akademie Verlag, Berlin, 1997.
  • Ree, R.: Lie elements and an algebra associated with shuffles. Ann. of Math. (2) 68 (1958), 210-220.
  • Reutenauer, C.: Free Lie algebras. London Mathematical Society Monographs, New Series 7. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1993.
  • Stroock, D. and Varadhan, S.: Multidimensional diffusion processes. Classics in Mathematics. Springer-Verlag, Berlin, 2006.