An Algebraic Correspondence Between Stochastic Differential Equations and the Martin–Siggia–Rose Formalism

In the realm of complex systems, dynamics is often modeled in terms of a nonlinear, 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.