We have seen in a previous article how the theory of “rough paths”
allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one
can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary,
we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals...

We show in this article how the theory of “rough paths”
allows us to construct solutions of differential
equations (SDEs) driven by processes generated by divergence-form
operators. For that, we use approximations
of the trajectories of the stochastic process by
piecewise smooth paths. A result of type Wong-Zakai
follows immediately.

We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus...

In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurable
coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these
quantities by solving some suitable
elliptic PDE problems.

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