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Exotic B-series representation of the Feller semigroup for Itô diffusions and the MSR path integral

Alberto Bonicelli

TL;DR

This work develops an exotic B-series framework for the Feller semigroup of one-dimensional Itô diffusions, bridging Itô-Taylor expansions with Martin-Siggia-Rose path-integral formalisms. It introduces exotic coloured trees, grafting rules, and a generalized Connes-Moscovici weight to obtain a convergent-like B-series representation of $\mathbb{E}^{u_0}[f(u_t)]$ with explicit combinatorial factors, while also addressing the asymptotic nature of the series. A parallel MSR perturbative program is formulated using Feynman multi-indices and diagrams, yielding an explicit MSR-Butcher-type expansion and a precise link to the exotic B-series via intertwiners $\Psi^{\mathfrak{e}}$ and $\Phi^{\mathfrak{e}}$. The results unify probabilistic and field-theoretic perspectives, show how contractions reproduce the time-power factors, and illustrate the approach with classical solvable diffusions, highlighting both the power and limitations of these perturbative constructions for stochastic dynamics.

Abstract

In this paper we consider the expansion of the Feller semigroup of a one-dimensional Itô 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.

Exotic B-series representation of the Feller semigroup for Itô diffusions and the MSR path integral

TL;DR

This work develops an exotic B-series framework for the Feller semigroup of one-dimensional Itô diffusions, bridging Itô-Taylor expansions with Martin-Siggia-Rose path-integral formalisms. It introduces exotic coloured trees, grafting rules, and a generalized Connes-Moscovici weight to obtain a convergent-like B-series representation of with explicit combinatorial factors, while also addressing the asymptotic nature of the series. A parallel MSR perturbative program is formulated using Feynman multi-indices and diagrams, yielding an explicit MSR-Butcher-type expansion and a precise link to the exotic B-series via intertwiners and . The results unify probabilistic and field-theoretic perspectives, show how contractions reproduce the time-power factors, and illustrate the approach with classical solvable diffusions, highlighting both the power and limitations of these perturbative constructions for stochastic dynamics.

Abstract

In this paper we consider the expansion of the Feller semigroup of a one-dimensional Itô 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.
Paper Structure (17 sections, 16 theorems, 156 equations)

This paper contains 17 sections, 16 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Exotic B-series expansion of expectations
  4. Itô-Taylor expansion and expectations
  5. Exotic coloured trees
  6. Grafting of decorated vertices
  7. Exotic B-series representation of the Feller semigroup
  8. Asymptotic nature of the series
  9. Realization map and the Connes-Moscovici weight
  10. MSR expectations via Feynman multi-indices
  11. MSR Feynman diagrams
  12. Feynman multi-indices
  13. Derivation of Equation \ref{['Eq: MSR Butcher']}
  14. Multi-indices exotic B-series
  15. Reduction of exotic trees and the exotic realization map
  16. ...and 2 more sections

Key Result

Proposition 1.1

Let $\alpha,\beta,f\in C^\infty(\mathbb{R})$ be real analytic functions such that Eq: SDE admits a unique solution $u$ with initial condition $u_0\in\mathbb{R}$ on $[0,T]$. Then the Feller semigroup associated to the solution of Eq: SDE admits the formal power series representation where the sum runs over all exotic coloured trees lying in ${\mathcal{T}}_{\alpha,\beta}^{\mathfrak{e}}$.

Theorems & Definitions (59)

  • Proposition 1.1
  • Proposition 1.2
  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 49 more