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October, 2009 A signed measure on rough paths associated to a PDE of high order: results and conjectures
Daniel Levin , Terry Lyons
Rev. Mat. Iberoamericana 25(3): 971-994 (October, 2009).


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.


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Daniel Levin . Terry Lyons . "A signed measure on rough paths associated to a PDE of high order: results and conjectures." Rev. Mat. Iberoamericana 25 (3) 971 - 994, October, 2009.


Published: October, 2009
First available in Project Euclid: 3 November 2009

zbMATH: 1193.60071
MathSciNet: MR2590691

Primary: 60H05 , 60H15

Keywords: Brownian motion , high-order PDE , rough path , Signature of a path

Rights: Copyright © 2009 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.25 • No. 3 • October, 2009
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