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A signed measure on rough paths associated to a PDE of high order: results and conjectures. (English) Zbl 1193.60071

As explained in the abstract of the paper, the authors extend the link between the Wiener measure and the Laplacian operator to high order differential operators of dissipative type and constant coefficients. Precisely, to any such an operator one can associate a signed measure on the space of continuous piecewise linear paths on \([0,T]\) through the corresponding continuous semigroup of operators.
Then, quoting from the introduction: “Fix a partition, and consider the expected signature for a piecewise linear path chosen randomly according to the measure associated to the partition and semigroup. Our main result is that, if we take the limit as the mesh of the partition goes to zero, this expected signature converges to a non-trivial limit which is readily calculated. The limit of the expected signature is an explicit tensor series in the tensor algebra.”
The proof of the main result of the paper makes use of analytic arguments, and these are essentially based on a suitable integration-by-parts formula. This has been important in order to check that the underlying differential operator remains the corresponding infinitesimal generator when considering a space of polynomials rather than the space of Schwartz functions.

MSC:

60H05 Stochastic integrals
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J65 Brownian motion
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