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

Abstract

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

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

Dates
First available in Project Euclid: 3 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1257258099

Mathematical Reviews number (MathSciNet)
MR2590691

Zentralblatt MATH identifier
1193.60071

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

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

Citation

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. https://projecteuclid.org/euclid.rmi/1257258099


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