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.