An algebraic correspondence between stochastic differential equations and the Martin-Siggia-Rose formalism
Alberto Bonicelli, Claudio Dappiaggi, Nicolò Drago
TL;DR
This work provides a rigorous perturbative bridge between stochastic differential equations with Gaussian white noise and the Martin-Siggia-Rose formalism by embedding both into an algebraic quantum field theory framework. Using deformed algebras (Gamma_G for MSR and Gamma_{\delta/2} for SDEs) and local polynomial functionals, the authors define expectation values and correlation functions in a way that makes the MSR path-integral more precise. They prove a perturbative SDE–MSR correspondence in two canonical cases: additive noise (beta = 1) and multiplicative noise (alpha = 0), showing that the AQFT-based expectations match the MSR expectations (with Itô/Stratonovich choices encoded by parameters). The results rely on detailed graph expansions and a careful treatment of cut-offs and renormalization-like structures, and point toward extensions to SPDEs and algebraic adiabatic limits. Overall, the paper offers a rigorous, perturbative framework for connecting stochastic dynamics and MSR field theories, with potential impact on rigor in stochastic modeling and renormalization in stochastic field theories.
Abstract
In the realm of complex systems, dynamics is often modeled in terms of a non-linear, stochastic, ordinary differential equation (SDE) with either an additive or a multiplicative Gaussian white noise. In addition to a well-established collection of results proving existence and uniqueness of the solutions, it is of particular relevance the explicit computation of expectation values and correlation functions, since they encode the key physical information of the system under investigation. A pragmatically efficient way to dig out these quantities consists of the Martin-Siggia-Rose (MSR) formalism which establishes a correspondence between a large class of SDEs and suitably constructed field theories formulated by means of a path integral approach. Despite the effectiveness of this duality, there is no corresponding, mathematically rigorous proof of such correspondence. We address this issue using techniques proper of the algebraic approach to quantum field theories which is known to provide a valuable framework to discuss rigorously the path integral formulation of field theories as well as the solution theory both of ordinary and of partial, stochastic differential equations. In particular, working in this framework, we establish rigorously, albeit at the level of perturbation theory, a correspondence between correlation functions and expectation values computed either in the SDE or in the MSR formalism.