  We study triangulation schemes for the joint kernel of a diffusion process
with uniformly continuous coefficients and an adapted, non-resonant Abelian
process. The prototypical example of Abelian process to which our methods apply
is given by stochastic integrals with uniformly continuous coeffcients. The
range of applicability includes also a broader class of processes of practical
relevance, such as the sup process and certain discrete time summations we
discuss. We discretize the space coordinate in uniform steps and assume that
time is either continuous or finely discretized as in a fully explicit Euler
method and the Courant condition is satisfied. We show that the Fourier
transform of the joint kernel of a diffusion and a stochastic integral
converges in a uniform graph norm associated to the Markov generator.
Convergence also implies smoothness properties for the Fourier transform of the
joint kernel. Stochastic integrals are straightforward to define for finite
triangulations and the convergence result gives a new and entirely constructive
way of defining stochastic integrals in the continuum. The method relies on a
reinterpretation and extension of the classic theorems by Feynman-Kac,
Girsanov, Ito and Cameron-Martin, which are also re-obtained. We make use of a
path-wise analysis without relying on a probabilistic interpretation. The
Fourier representation is needed to regularize the hypo-elliptic character of
the joint process of a diffusion and an adapted stochastic integral. The
argument extends as long as the Fourier analysis framework can be generalized.
This condition leads to the notion of non-resonant Abelian process.
