Representation of the semigroup for Itô diffusions via (multi-indices) exotic B-series and Feynman diagrams

In this talk, after justifying the expansion of the semigroup of a one-dimensional Itô diffusion as a power series in time, I will build on previous results on expansions labelled by exotic rooted trees to derive an explicit expression for the combinatorial factors involved. A key step is the extension of the notion of tree factorial and Connes-Moscovici weights to this richer family of rooted trees. As a reult, we obtain an exotic Butcher series representation of the semigroup, suitable for a comparison with the perturbative path integral construction of the statistics of the diffusion, known in the literature as Martin-Siggia-Rose formalism. Computations in the latter framework are based on the erroneous assumption that the measure of the path integral can be seen as a perturbation of a Gaussian measure. Resorting to multi-indices to represent pre-Feynman diagrams, I will shed some light on why, even if starting from such an assumption, the results happen to be correct.