Exotic B-series representation of the Feller semigroup for It\^o diffusions and the MSR path integral

TL;DR

This paper develops an exotic B-series expansion for the Feller semigroup of one-dimensional Itô diffusions, connecting combinatorial tree structures with path integral formalisms.

Contribution

It introduces a novel exotic B-series representation for the Feller semigroup, extending tree factorials and weights to richer rooted trees, and links it to the MSR path integral formalism.

Findings

Derived explicit combinatorial factors for the exotic B-series.

Extended tree factorial and Connes-Moscovici weight concepts.

Established the equivalence between the exotic B-series and the MSR path integral.

Abstract

In this paper we consider the expansion of the Feller semigroup of a one-dimensional It\^o diffusion as a power series in time. Taking our moves from previous results on expansions labelled by exotic trees, we derive an explicit expression for the combinatorial factors involved, that leads to an exotic Butcher series representation. A key step is the extension of the notion of tree factorial and Connes-Moscovici weight to this richer family of rooted trees. The ensuing expression is 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. Resorting to multi-indices to represent pre-Feynman diagrams, we show that the latter coincides with the exotic B-series representation of the semigroup, giving it a solid mathematical foundation.